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

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 X22 + υt

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.

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.