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Slide 11.1 Undergraduate Econometrics,2 nd   Edition-Chapter 11 Chapter 11Heteroskedasticity11.1The Nature of Heteroskedasticity ã   In Chapter 3 we introduced the linear model 12  yx =β +β  (11.1.1)to explain household expenditure on food (  y ) as a function of household income (  x ). ã   We begin this section by asking whether a function such as  y  = β 1  + β 2  x  is better atexplaining expenditure on food for low-income households than it is for high-incomehouseholds. ã   Income is less important as an explanatory variable for food expenditure of high-income families. It is harder to guess their food expenditure.  Slide 11.2 Undergraduate Econometrics,2 nd   Edition-Chapter 11 ã   This type of effect can be captured by a statistical model that exhibitsheteroskedasticity. 12 ttt   yxe =β +β +  (11.1.2) ã   We assumed the e t    were uncorrelated random error terms with mean zero and constantvariance σ 2 . That is,()0 t   Ee  =   2 var() t  e  = σ  cov(,)0 ij ee  =  (11.1.3) ã   Including the standard errors for b 1  and b 2 , the estimated mean function wasˆ t   y  = 40.768+0.1283 t   x  (11.1.4) (22.139)(0.0305) ã   A graph of this estimated function, along with all the observed expenditure-income points (,) tt   yx , appears in Figure 11.1.  Slide 11.3 Undergraduate Econometrics,2 nd   Edition-Chapter 11 ã    Notice that, as income (  x t  ) grows, the observed data points (,) tt   yx  have a tendency todeviate more and more from the estimated mean function. ã   The least squares residuals, defined by 12 ˆ ttt  eybbx = − −  (11.1.5)increase in absolute value as income grows. [Figure 11.1 here] ã   The observable least squares residuals ˆ() t  e  are proxies for the unobservable errors () t  e that are given by 12 ttt  eyx = −β −β  (11.1.6) ã   The information in Figure 11.1 suggests that the unobservable errors also increase inabsolute value as income () t   x  increases. ã   Is this type of behavior consistent with the assumptions of our model?  Slide 11.4 Undergraduate Econometrics,2 nd   Edition-Chapter 11 ã   The parameter that controls the spread of  y t   around the mean function, and measures theuncertainty in the regression model, is the variance σ 2 . ã   If the scatter of t   y  around the mean function increases as  x t   increases, then theuncertainty about t   y  increases as t   x  increases, and we have evidence to suggest thatthe variance is not constant. ã   Thus, we are questioning the constant variance assumption 2 var()var() tt   ye = = σ  (11.1.7) ã   The most general way to relax this assumption is to add a subscript t   to σ 2 , recognizingthat the variance can be different for different observations. We then have 2 var()var() ttt   ye = = σ  (11.1.8) ã   In this case, when the variances for all observations are not the same, we say that heteroskedasticity  exists. Alternatively, we say the random variable t   y and the randomerror t  e  are heteroskedastic .

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Jul 23, 2017

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