Cross-sectional:Observations on individuals, households, enterprises, countries, etc at one moment in time (Chapters 1–10, Models A and B). 1 During this course we will work with the three types of data described above. TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A TYPES OF DATA Time series:Observations on income, consumption, interest rates, etc over a number of time periods (years, quarters, months, …) (Chapters 11–13, Model C). Panel data:Observations on the same cross-section of individuals, households, etc over a number of time periods (Chapter 14, Model B).

Different regression models are appropriate for different types of data. We will consider three types of regression model, as shown above. 2 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A TYPES OF MODEL Model A:Cross-sectional data with nonstochastic regressors. Their values in the observations in a sample do not have stochastic (random) components. Model C:Time series data. The values of the regressors may exhibit persistence over time. Regressions with time series data potentially involve complex technical issues that are best avoided initially. Model B:Cross-sectional data with stochastic regressors. The values of the regressors are drawn randomly and independently from defined populations.

We will start with Model A. We will do this purely for analytical convenience. It enables us to conduct the discussion of regression analysis within the relatively straightforward framework of what is known as the Classical Linear Regression Model. 3 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A TYPES OF MODEL Model A:Cross-sectional data with nonstochastic regressors. Their values in the observations in a sample do not have stochastic (random) components. Model C:Time series data. The values of the regressors may exhibit persistence over time. Regressions with time series data potentially involve complex technical issues that are best avoided initially. Model B:Cross-sectional data with stochastic regressors. The values of the regressors are drawn randomly and independently from defined populations.

We will replace it in Chapter 8 by the weaker and more realistic assumption, appropriate for regressions with cross-sectional data, that the observations on the regressors are randomly drawn from defined populations. 4 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A TYPES OF MODEL Model A:Cross-sectional data with nonstochastic regressors. Their values in the observations in a sample do not have stochastic (random) components. Model C:Time series data. The values of the regressors may exhibit persistence over time. Regressions with time series data potentially involve complex technical issues that are best avoided initially. Model B:Cross-sectional data with stochastic regressors. The values of the regressors are drawn randomly and independently from defined populations.

A.1The model is linear in parameters and correctly specified. ‘Linear in parameters’ means that each term on the right side includes a  as a simple factor and there is no built-in relationship among the  s. We will defer a discussion of issues relating to linearity and nonlinearity to Chapter 4. 5 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A For example: Examples of models that are not linear in parameters:

There must be some variation in the regressor in the sample. Otherwise it cannot account for any of the variation in Y. 6 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.2There is some variation in the regressor in the sample.

If we tried to regress Y on X, when X is constant, we would find that we would not be able to compute the regression coefficients. Both the numerator and the denominator of the expression for b 2 would be equal to zero. We would not be able to obtain b 1 either. 7 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A If for all i, A.2There is some variation in the regressor in the sample. ASSUMPTIONS FOR MODEL A

8 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A We assume that the expected value of the disturbance term in any observation should be zero. Sometimes the disturbance term will be positive, sometimes negative, but it should not have a systematic tendency in either direction. A.3The disturbance term has zero expectation ASSUMPTIONS FOR MODEL A for all i

9 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A Actually, if an intercept is included in the regression equation, it is usually reasonable to assume that this condition is satisfied automatically. The role of the intercept is to pick up any systematic but constant tendency in Y not accounted for by the regressor(s). ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i

10 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A Suppose that the disturbance term had a nonzero population mean. ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i Suppose

11 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A Define a new random variable v i = u i –  u. ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i Define Suppose

12 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A Then we can rewrite the model as shown. v i becomes the new disturbance term and the intercept has absorbed the constant  u. ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i Define Suppose where

13 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A The disturbance term in the revised model now satisfies Assumption A.3. ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i Define Suppose where Then

The price that we pay is that the interpretation of the intercept has changed. It has absorbed the nonzero component of the disturbance term in addition to whatever had previously been responsible for it. 14 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation for all i Define Suppose where Then

15 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A This is usually acceptable because the role of the constant is usually to pick up any systematic tendency in Y not accounted for by the regressor(s). for all i Define Suppose ASSUMPTIONS FOR MODEL A A.3The disturbance term has zero expectation where Then

We assume that the disturbance term is homoscedastic, meaning that its value in each observation is drawn from a distribution with constant population variance. 16 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A for all i A.4The disturbance term is homoscedastic ASSUMPTIONS FOR MODEL A

In the language of the section on sampling and estimators in the Review chapter, this is a ‘beforehand’ concept, where we are thinking about the potential behavior of the disturbance term before the sample is actually generated. 17 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.4The disturbance term is homoscedastic for all i

Once we have generated the sample, the disturbance term will turn out to be greater in some observations, and smaller in others, but there should not be any reason for it to be more erratic in some observations than in others. 18 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.4The disturbance term is homoscedastic for all i

Since E(u i ) = 0, by Assumption A.3, the population variance of u i is equal to E(u i 2 ), so the condition can also be written as shown. 19 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A for all i ASSUMPTIONS FOR MODEL A A.4The disturbance term is homoscedastic for all i

If Assumption A.4 is not satisfied, the OLS regression coefficients will be inefficient, and you should be able to obtain more reliable results by using a modification of the regression technique. This will be discussed in Chapter TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.4The disturbance term is homoscedastic for all i

We assume that the disturbance term is not subject to autocorrelation, meaning that there should be no systematic association between its values in any two observations. 21 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A A.5The values of the disturbance term have independent distributions ASSUMPTIONS FOR MODEL A u i is distributed independently of u j for all j ≠ i

For example, just because the disturbance term is large and positive in one observation, there should be no tendency for it to be large and positive in the next (or large and negative, for that matter, or small and positive, or small and negative). 22 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.5The values of the disturbance term have independent distributions u i is distributed independently of u j for all j ≠ i

The assumption implies that the population covariance between u i and u j is zero. Note that the population means of u i and u j are both zero, by virtue of Assumption A.3, and that E(u i u j ) can be decomposed as E(u i )E(u j ) if u i and u j are generated independently – see the Review. 23 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A u i is distributed independently of u j for all j ≠ i ASSUMPTIONS FOR MODEL A A.5The values of the disturbance term have independent distributions

If this assumption is not satisfied, OLS will again give inefficient estimates. Chapter 12 discusses the problems that arise and ways of getting around them. Violations of this assumption are in any case rare with cross-sectional data. 24 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A ASSUMPTIONS FOR MODEL A A.5The values of the disturbance term have independent distributions u i is distributed independently of u j for all j ≠ i

We usually assume that the disturbance term has a normal distribution. The justification for the assumption depends on the Central Limit theorem. 25 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A A.6The disturbance term has a normal distribution ASSUMPTIONS FOR MODEL A

26 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A In essence, the CLT states that, if a random variable is the composite result of the effects of a large number of other random variables, it will have an approximately normal distribution even if its components do not, provided that none of them is dominant. ASSUMPTIONS FOR MODEL A A.6The disturbance term has a normal distribution

27 TYPES OF REGRESSION MODEL AND ASSUMPTIONS FOR MODEL A The disturbance term u is composed of a number of factors not appearing explicitly in the regression equation so, even if we know nothing about the distribution of these factors, we are usually entitled to assume that the disturbance term is normally distributed. ASSUMPTIONS FOR MODEL A A.6The disturbance term has a normal distribution

Copyright Christopher Dougherty 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 2.2 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 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 or the University of London International Programmes distance learning course ec2020 Elements of Econometrics