ECON301_Handout_02_1213_02

Figure 4 consumption expenditures for different

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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 Eu ; t =1,2,3,…, T c) “constant variance”: 2 () t Var u ; t =1,2,3,…, T d) “zero covariance”: ts Cov(u ,u ) 0 ; st ; s,t =1,2,3,…, T . or “independence”: u ,u independent; ; 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 uN ; 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 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 ). If the regressors in X are nonrandom and the disturbances are 2 (0, ) IID , we do not assume normality of the disturbance distributions, it can still be argued that the distributions of OLS estimators ( ˆ i ) is approximately normal provided that the sample size is reasonably large. 3. Sampling Distribution and Repeated Sampling Suppose we perform the experiment of taking what is called a repeated sample: keeping the values of the independent variables unchanged, we obtain new observations for the dependent variable by drawing a new set of disturbances. This could be repeated, ay, 5000 times, obtaining 5000 of these repeated samples. For each of these
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Figure 4 Consumption Expenditures for Different Levels of...

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