Documents

Undergraduate Econometric

Description
Chapter 11 Slides
Categories
Published
of 32
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  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 .
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks