short notes on Autocorrelation in Econometrics. Muhammad Ali Lecturer in Statistics Higher Education Department, KPK, Pakistan.

1. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
1
Autocorrelation
Definition
The classical assumptions in the linear regression are that the errors terms i have zero mean and
constant variance and are uncorrelated [E( i) = 0, Var( i) = δ2
, and E( i j ) = 0 ]. For the
construction of Confidence Interval, and Testing of hypothesis about the regression coefficients
we add the assumption of normality. so that i are NID(0, δ2
). Some applications of regression
involve regressor and response variables that have a natural sequential order over time. Such data
are called time series data. Regression models using time series data occur relatively often in
economics, business, and some fields of engineering. The assumption of uncorrelated or
independent errors for time series data is often not appropriate. Usually the errors in time series
data exhibit serial correlation, that is, E( i j ) ≠ 0. Such error terms are said to be
autocorrelated. Autocorrelation sometimes called "lagged correlation or "serial correlation".
Causes of Autocorrelation
Specification Bias:
a) Excluded Variables Case
There are several causes of autocorrelation. Perhaps the primary cause of
autocorrelation in regression problems involving time series data is failure to include one
or more important regressors in the model. For example suppose that we wish to regress

2. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
2
annual sales of a soft drink company against the annual advertising expenditure for that
product. Now the growth in population over the period of time used in the study will also
influence the product sales. If population size is not included in the model, this may cause
the errors in the model to be positively autocorrelated, because population size is
positively correlated with product sales.
Consider the true model:
Sale (Yt) = β0 + β1X1t + β2X2t + εt ---------------------- ( I )
Where Y is the sale, X1 is the advertising expenditure, X2 is the population size.
However for some reason we run the following regression:
Sale (Yt) = β0 + β1X1t + υt ---------------------- ( II )
As model ( I ) is a true model and we run model ( II ), and hence the error or disturbance
term υ will be autocorrelated.
b) Incorrect Functional Form:
Consider the following cost and output model:
Yt = β1 + β2 X1 + β3 X2
2
+ υt

3. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
3
Instead of using the above form which is considered to be correct, if we fit the
following model:
Yt = β1 + β2 X1 + β3 X2 + υt
In this case, υ will reflect autocorrelation because of the use of an incorrect
functional form.
Theoretical consequences of autocorrelation
The presence of autocorrelation in the errors has several effects on the ordinary least-squares
regression procedures. These are summarized as follows:
1. Ordinary least-squares regression coefficients are still unbiased.
2. OLS regression coefficients are no longer efficient i..e. they are no longer minimum
variance estimates. We say that these estimates are inefficient.
3. The residual mean square MSres may seriously underestimate δ2
. Consequently, the
standard errors of the regression coefficients may be too small. Thus, confidence intervals
are shorter than they really should be, and tests of hypothesis on individual regression
coefficients may indicate that one or more regression contribute significantly to the
model when they really do not. Generally, underestimating δ2
gives the researcher a false
impression of accuracy.
4. The confidence intervals and tests of hypothesis based on the t and F distributions are no
longer appropriate.

4. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
4
OLS estimates in presence of autocorrelation
There are three main consequences of autocorrelation on the ordinary least squares estimates.
1. Ordinary least squares regression coefficients are still unbiased even if the disturbance
term is autocorrelated. i.e.
We know that
( )
( ) ( )
( )
εβ
εβ
εβ
εβεβ
β
XXX
XXXI
XXXXXXX
XYXXXX
YXXX
′′+=
′′+=
′′+′′=
+=∴+′′=
′′=
−
−
−−
−
−
1
1
11
1
1
)(
)(
)()(
ˆ
Taking expectation on both sides of the above equation #1, assuming that E(ε) = 0 i.e.
β
β
εββ
=
+=
′+= −
0
)()()ˆ( 1
XEXXE
Hence in the presence of autocorrelation the OLS estimates are still unbiased.