Chapter 12. Autocorrelation

1. Autocorrelation = Serial correlation = situation when the errors in different equations are correlated.

Using model (12.12) and Figure 12.1 explain the problem arising from autocorrelation (biased estimates) and how it is bypassed in large samples (consistency). Describe positive and negative autocorrelation.

The consequences are the same as of heteroscedasticity (because both are deviations from ): unbiasedness remains but efficiency is lost. The standard errors are estimated wrongly. Positive autocorrelation is common in time series where successive observations are not independent. Negative autocorrelation is uncommon and usually arise when the series are differenced.

1. Detection of first-order autocorrelation:

The Breusch_Godfrey test

The Durbin-Watson test

Consider

(1)

where the error is AR(1)

(2) .

Here is a new error, called an innovation. Innovations are assumed to be white noise: .

The Durbin-Watson statistic

is calculated from the residuals and in large samples

.

Describe the relation between and geometrically and explain the use of the lower and upper values .

Explain how in case of the model (1) + (2) nonlinear least squares can be used to estimate the coefficients, including .

1. Fitting a model subject to AR(1) autocorrelation
2. The common factor test

Explain how an attempt to rewrite (1) + (2) in a form free from autocorrelation motivates the common factor test. How would you use NLS with restrictions to perform this test?

1. Apparent autocorrelation

Using models (12.47) and (12.48) explain how a wrong model specification may lead to apparent autocorrelation.

1. Model specification: specific-to-general versus general-to-specific

A model is said to be nested inside another if it can be obtained from it by imposing a number of restrictions. Two models are said to be non-nested if neither can be represented as a restricted version of the other.

Exercise. What is the relationship between the models:

(A)

(B)

(C)

(D) ?

If the rival models are not nested, one can create a union model embracing the two models as restricted versions and test each rival against the union.

Example. Suppose the rivals models are

(E)

(F)

Then the union model is

.

It may not have economic sense, and may be highly correlated.

Instead of going from simple to more complex models by including new variables, one can go from the general to the specific. The problem is that in a large model many variables may be correlated and have insignificant coefficients and, as a result, many will have to be excluded.

Exercises. 12.1, 12.2, 12.3, 12.7, 12.9, 12.11

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