Serial Correlation

Serial correlation is a problem associated with time-seriesdata.

It occurs when the errors of the regression model arecorrelated with their own past values.

Serial correlation by itself does not mean that OLS willbe biased.

The most common effect of serial correlation is to biasdownwards the standard errors of the OLS estimator(though this is not always the case).

t t tY X u

0 for some 0t t kE u u k

Definition of Serial Correlation

Consider the regression model

The errors of this model are said to be serially correlated if

Since this is not consistent with the second Gauss-Markovassumption, it follows that OLS is no longer BLUE.

Serial correlation and autocorrelation

Serial correlation is a general term used to describe anysituation in which the error terms are not completely independent.

Autocorrelation is a particular type of serial correlation in whichthe error terms are a function of their own past values.

1

1

t t t

t t t

u u

u

Both of these equations describe serial correlation but only thefirst describes autocorrelation. The second equation describes amoving average error which is a different type of serial correlation.

Orders of serial correlation

In general the order of serial correlation refers to the maximumlag on the right hand side of the equation describing the errorterm. For example:

1 1 2 2 3 3 4 4t t t t t tu u u u u

is a fourth-order autocorrelation process.

The order of the serial correlation process is often related to thefrequency of the data. For example, models estimated with quarterly data often have errors which exhibit fourth-order autocorrelation.

Suppose we have a regression model of the form

t t tY X u

We wish to test for the presence of serial correlationin the residuals which are defined as:

ˆˆt t tu Y X

Detection of serial correlation

Formal methods for the detection of serial correlation arebased on the sample autocorrelations. These are defined as:

2

ˆ ˆˆ

ˆt t k

kt

u u

u

The sample correlogram gives a visual guide to the structureof the sample autocorrelations. This can be used as a diagnostictool for a regression model.

The shape of the correlogram depends on the type of serialcorrelation. For example consider the case of first order positive autocorrelation:

10.75t t tu u

i 0 10

0 2 4 6 8 100

0.5

1

i

i

The Durbin-Watson test

The Durbin-Watson test provides a more formal test for thepresence of first-order autocorrelation.

Consider the following model:

1

t t t

t t t

Y X u

u u

We wish to test:

0

1

: 0

: 0

H

H

Consider the test statistic

2

1

2

2 21 1

2

ˆ ˆ

ˆ

ˆ ˆ ˆ ˆ2

ˆ

t t

t

t t t t

t

u uDW

u

u u u u

u

If the sample size is large then

2 21ˆ ˆt tu u

and therefore

1

2

ˆ ˆˆ2 1 2 1

ˆt t

t

u uDW

u

Under the null hypothesis that there is no autocorrelation wehave

E(DW)=2

If there is positive autocorrelation then E(DW)<2.

If there is negative autocorrelation then E(DW)>2

Note that DW is bounded between 0 and 4.

Interpreting the Durbin-Watson Test

0 42d d 4-d4-dL U U L

Accept null hypothesis

Reject null hypothesis

Uncertain