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Unformatted text preview: Lecture 9 Heteroskedasticity In this chapter, we aim to answer the following questions: 1. What is the nature of heteroskedasticity? 2. What are its consequences? 3. how does one detect it? 4. What are the remedial measures? 9.1 The Nature of Heteroskedasticity • Homoskedasticity (homoskedasticity): the classical regression model assumes that residuals ε i were identically distributed with mean zero and equal variance σ 2 (i.e., E( ε i  X i ) = 0, and Var( ε i  X i ) = σ 2 , where X i means { X i 2 , ··· , X ik } , for i = 1 , 2 , ··· , n ) • Because the variance is a measure of dispersion of the observed value of the dependent variable ( y around the regression line β 1 + β 2 X 2 + ··· + β k X k ), homoskedasticity means that the dispersion is the same across all observations. However, in many situations, this assumption might be false. Example 1. Take the sample of household consumption expenditure and income. Since household with low income do not have much flexi bility in spending, consumption patterns among such lowincome house holds may not vary very much. On the other hand, rich families have 1 2 LECTURE 9 HETEROSKEDASTICITY a great deal of flexibility in spending. Some might be large consumers; others might be large savers and investors in financial markets. This implies that actual consumption might be quite different from average consumption. Therefore, it is very likely that higher income households have a large dispersion around mean consumption than lower income households. Such situation is called heteroskedasticity. Example 2. The annual salary and the number of years since earning the Ph.D. for 222 professors from seven universities. (Example 8.1, Ramanathan (2002)) Look at the scatter diagram of log salary and years since Ph.D. [Figure 8.2] The spread around an average straightline relation is not uniform. ⇒ It violates the usual assumption of homoskedasticity of the error terms. • Heteroskedasticity arises also when one uses grouped data rather than individual data. • Heteroskedasticity can occur in time series data also. • Let’s relax the assumption that the residual variance is constant across observations and assume heteroskedasticity instead • Assume ε i is a random variable with E( ε i  X i ) = 0 and Var( ε i  X t ) = E ( ε 2 i  X i ) = σ 2 i , for i = 1 , ··· , n . It implies each observation has a different error variance . ASSUMPTION A4’ ε i is a random variable with E( ε i  X i ) = 0 and Var( ε i  X i ) = E ( ε 2 i  X i ) = σ 2 i , for i = 1 , ··· , n . Thus, E[ εε ] = σ 2 Ω = σ 2 1 ··· σ 2 2 ··· . . . . . . . . . . . . . . . ··· σ 2 n It will sometimes be useful to write σ 2 i = σ 2 ω i For convenience, we shall use the normalization tr( Ω ) = n X i =1 ω i = n c YinFeng Gau 2002 ECONOMETRICS 3 LECTURE 9 HETEROSKEDASTICITY 9.2 Consequences of Ignoring Heteroskedas ticity y i = β 1 + β 2...
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 Spring '11
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