Detail notes on Autocorrelation, including all mathematical work. 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 ------------------- ( III )

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 ----------------( IV)
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.

5. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
5
2. The residual mean square underestimate δ2
result in small standard errors of the
regression coefficients.
We know that variance of the OLS estimate is:
)(ˆ
Putˆ
(B)equationinvaluesthese
1and0
)(
/
)(
)(
;
ereWh;
0)(X
)(
)(
)(
)()(
)(
))((ˆ
)(]ˆ[)ˆ(
11
2
2
10
10
2
2
i2
22
2
cw
w
Putting
Xw
x
XX
xxwSince
BwXww
Xw
x
x
wYw
XXx
x
xY
X
XX
XXY
XX
XXYXXY
XX
YYXX
Since
AEVar
ii
ii
ii
i
i
iii
iiiii
iii
i
i
iii
ii
i
ii
i
ii
i
iii
i
ii
−−−−−−−−∑=−
=∑+=
=∑=
∑
−∑
=∑∑=∑
−−−−−−−∑+∑+∑=
++∑=
∑
=∴∑=
−=
∑
∑
=
=−∑∴
−∑
−∑
=
−∑
−∑−−∑
=
−∑
−−∑
=
−−−−−−=
εββ
ββεββ
εββ
εββ
β
βββ

6. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
6
Putting the values of equation ( c) in equation (A).
[ ] [ ]
[ ]
[ ]
∑ ∑= <
−−
−−
+=
+++++++=
+++++++=
+++=∑=
n
i
n
ji
jijiii
nnnnnn
nnnnnn
iniiii
wwEwE
wwwwwwwwwE
wwwwwwwwwE
wwwEwEVar
1
22
1131312121
222
2
2
2
2
1
2
1
1131312121
222
2
2
2
2
1
2
1
2
22
2
][2][
...[2]...
2...22...
...)ˆ(
εεε
εεεεεεεεε
εεεεεεεεε
εεεεβ
∑ ∑= <
−−−−−−+=
n
i
n
ji
jijiii DEwwEw
1
22
)()(2)( εεε
If there is no correlation between error terms i.e. E( 0) =jiεε then equation (D)
becomes:
( )
)(
)(
/
)(
0)()ˆ(
2
2
22
22
2
2
2
2
222
22
1
22
E
XX
x
x
x
x
x
w
wEwVar
i
i
i
i
i
i
i
i
n
i
ii
−−−−−
−∑
=∑=
∑
∑
=
∑
∑=∑=
∑=+= ∑=
δ
δδ
δδ
δεβ
Now we have to find ( )βˆVar when the errors are autocorrected. i.e. errors are AR(1).

7. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
7
Under AR (1)
= + (0,
( = ( + ( = 0
( = ( + (
= +
− =
=
1 −
As ( = ( = ( = ( =
Now ( = ( + ( = ( + ( =
!
"!
#$%&'(1 = )
∑ +
∑ +
, = )
∑ + + 2∑+ +
(∑ +
, =
∑ +
+
2∑+ +
(∑ + 1 −
=
1
∑ + 1 −
+
2∑+ +
(∑ + 1 −
=
1 −
1
∑ +
)1 +
2∑+ +
∑ +
, − − − − − (.
It is clear from equation (E) and (F) that 1)ˆ()ˆ( 22 ARVarVar ββ < . Therefore, if we use )ˆ( 2βVar ,
we shall inflate the precision or accuracy of the estimator 2
ˆβ . As a result, the t ratio will be
overestimated.
Methods of detection of Autocorrelation
Following are the methods of detecting the problem of autocorrelation:
1. Residual plots: Residuals plots can be useful for the detection of autocorrelation. The
most meaningful display is the plot of residuals versus time. If there is positive

8. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
8
autocorrelation, residuals of identical sign occur in clusters. That is, there are not
enough changes of sign in the pattern of residuals. On the other hand, if there is negative
autocorrelation, the residuals will alternate signs too rapidly.
2. The Runs Test: The Run test having the following steps.
Step1. Write down the plus and minus sign's of the residuals.
Step2. Count the number of plus signs, negative signs and total number of runs( A run is
a sequence of either positive or negative signs without interruption). Now let
N= Total number of observations= N1+N2
N1=Number of + symbols (i.e. + residuals)
N2=Number of - symbols (i.e. - residuals)
R= number of runs
Step3. Compare the value of 'R' with that of the tabulated value, if it is less than the
smaller tabulated value or greater that the larger tabulated value then we have to reject
the hypothesis that pattern of errors are random i.e. H0; The sequence of errors are
random. In other words, the residuals exhibit autocorrelation.
Special case: If N1 > 20 or N2 > 20 or both then the number of runs is asymptotically
normally distributed with
Mean: 1
2 21
+=
N
NN
Rµ & Variance:
)1()(
)2(2
2
21212
−
−
=
NN
NNNNN
Rδ
Testing of hypothesis procedure is the same as of the Z-test.
Note that the run test sometimes also known as the Geary test, a nonparametric test.

9. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
9
3. Duban-Watson d Test: This test is based on the assumption that the errors in the
regression model are generated by a first-order autoregressive process observed at
equally spaced time periods, i.e.
= + --------------( i )
where is the error term in the model at time t, is an NID(0,/2
ε) random variable,
and ρ is the autocorrelation parameter. Thus, a simple linear regression model with first-
order autoregressive errors would be
yt = β0 + β1 xt + εt---------------( ii )
Where yt and xt are the observations on the response and regressor variables at time
period t. The hypothesis usually considered in the Dubrin-Watson test are
Ho: ρ = 0
H1: ρ > 0
The test statistic is
( )
2
2
2
1
t
n
t
tt
e
ee
d
∑
−
=
∑=
−
The value of the d statistic lies between two bounds, say dL and dU, such that if d is
outside these limits, a conclusion regarding the hypothesis can be reached. The decision
procedure is as follows:
Reject H0 if:
0 ≤ d ≤ dL Evidence of positive autocorrelation
4−dL ≤ d ≤ 4 Evidence of negative autocorrelation

10. Muhammad Ali
Lecturer in Statistics
GPGC Mardan.
10
Do not reject H0 if:
dU ≤ d ≤ 4−dU
Zone of indecision if:
dL ≤ d ≤ dU or 4−dU ≤ d ≤ 4− dL
Values of dL and dU can be obtained from the Durban-Watson table.
Situation where negative autocorrelation occurs are not often encountered. However, if a
test for negative autocorrelation is desired, one can use the statistic 4−d, where d is
defined above, then the decision rules for H0 : ρ = 0 versus H1 : ρ < 0 are the same as
those used in testing for positive autocorrelation. It is also possible to conduct a two-
sided ( H0 : ρ = 0 versus H1 : ρ ≠ 0 ) by using both one-sided tests simultaneously. If this
is done, the two-sided procedure has Type I error 2α, where α is the Type I error used for
each one-sided test.