780 980 880 100 100 0 100 0 2 100 0 2 2 10000 1400

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780, 980 880 -100, 100 0 [(-100-0) 2 +(100-0) 2 ]/2= 10000 1400 780, 850, 900, 930, 1240 940 -160, -90, -40, -10, 300 0 [(-160-0) 2 +(-90-0) 2 +(-40-0) 2 + (-10-0) 2 +(300-0) 2 ]/5= 25080 Like before, this figure shows the distribution of consumptions for four different income levels. Unlike before, these distributions are different from each other. Because the variance of the distribution of u t , given X t , ( | ) 0 t tt Var u X , depends on X t , u t is heteroscedastic. Figure 4 Consumption Expenditures for Different Levels of Disposable Income (Population) Heteroscedasticity Case
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 10 Figure 5 Probability Mass Functions for Y at Different Levels of Income (Heteroscedasticity Case) Note that if the distribution of Y were normal instead of discrete uniform as in our example, the graph above would have the form given below. Figure 6 Probability Density Functions for Y at Different Levels of Income (Heteroscedasticity Case)
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 11 2. Gauss-Markov Assumptions Assumptions 2, 3, 4 and 5 are of special importance. They are called Gauss-Markov assumptions . Hence, Gauss-Markov assumptions are as follows: a) “fixed X”: X t is nonrandom, t =1,2,3,…, T b) “zero mean”: ( ) 0 t E u ; t =1,2,3,…, T c) “constant variance”: 2 ( ) t Var u ; t =1,2,3,…, T d) “zero covariance”: t s Cov(u ,u ) 0 ; s t ; s,t =1,2,3,…, T . o r “independence”: t s u ,u independent; s t ; s,t =1,2,3,…, T . Very Important! To be able to carry out hypothesis testing about the estimated parameters (i.e. for inference) in case of small samples (n<30), we further assume that the disturbance term t u has a normal distribution: e) “normality”: ( ) t u N ; t =1,2,3,…, T . To ensure that the normality assumption is correctly stated, we can combine assumptions (b), (c) the independence version of (d), and (e), by writing 2 (0, ) t u NID . This means that under these conditions t u is normally and independently distributed ( NID ) with ( ) 0 t E u and 2 ( ) t Var u . If we do not wish to assume normality (since this is a strong assumption), we can assume that 2 (0, ) t u IID ; this means that the t u are independently and identically distributed and includes (b), (c), the independent version of (d), and the further assumption that all t u distributions are identical, including all moments not just the mean and variance [at least four moments as stated in Stock and Watson (2003, p.106)]. The modern approach to econometrics
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ECON 301 (01) - Introduction to Econometrics I March, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 12 drops the normality assumption and simply assumes that t u are independently draws from an identical distribution ( IID ).
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