1

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

Assumptions C.1, C.3, C.4, C.5, and C.8, and the consequences of their violations are the same as those for Model B and they will not be discussed further here.

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

Assumption C.2 replaces the Model B assumption that the values of the regressors in the observations in the sample are drawn randomly from fixed populations.

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ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

The Model B assumption is wholly unrealistic in a time series setting because the values of many time series variables are correlated, often strongly so, from one time period to the next.

3

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

A discussion of the meaning of ‘weakly persistent’ and the importance of this assumption leads us to relatively advanced technical issues and will be deferred to Chapter 13.

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ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

Assumption C.6 is rarely an issue with cross-sectional data. When observations are generated randomly, there is no reason to suppose that there should be any connection between the value of the disturbance term in one observation and its value in any other.

5

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

ut is distributed independently of ut' for t' ≠ t

However, with time series data it often happens that the value of the disturbance term in one observation is correlated with its value in the next. The reasons for this and its consequences are discussed in the next slideshow but one.

6

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

ut is distributed independently of ut' for t' ≠ t

Assumption C.7, like Assumption B.7, requires that the disturbance term be distributed of the values of all of the regressors in all of the observations.

7

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

ut is distributed independently of Xjt' for all t' (including t) and

j

C.2 The time series for the regressors are weakly persistent.

Assumption C.7, like Assumption B.7, requires that the disturbance term be distributed of the values of all of the regressors in all of the observations.

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ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7' The disturbance term has zero conditional expectation

C.2 The time series for the regressors are weakly persistent.

E(ut │values of all the regressors in all observations) = 0

We have already seen, in the context of cross-sectional data, how violations of Assumption B.7 can lead to inconsistent estimates (measurement error bias in Chapter 8 and simultaneous equations bias in Chapter 9).

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ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

The assumption is of even greater practical importance in the context of time series regressions. It will be discussed in detail in the next slideshow.

10

ASSUMPTIONS FOR MODEL C: REGRESSIONS WITH TIME SERIES DATA

C.1 The model is linear in parameters and correctly specified.

Y = b1 + b2X2 + … + bkXk + u

C.2 The time series for the regressors are weakly persistent.

C.3 There does not exist an exact linear relationship among the regressors.

C.4 The disturbance term has zero expectation.

C.5 The disturbance term is homoscedastic.

C.6 The values of the disturbance term have independent distributions

C.7 The disturbance term is distributed independently of the regressors

C.8 The disturbance term has a normal distribution.

Copyright Christopher Dougherty 2013.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 11.1 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own who feel that they might benefit

from participation in a formal course should consider the London School of

Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2013.01.20