The name changes to probability density function when

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The name changes to probability density function when the distribution is continuous such as normal distribution .
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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 6 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 3 Probability Density Functions for Y at Different Levels of Income (Homoscedasticity Case) Important! Thus, the assumption that the Population Regression Function (PRF) passes through the conditional means of Y t implies that the conditional mean values of t u (conditional on the given X’s) are zero. In other words, the assumption that ( | ) 0 t t E u X is equivalent to assuming that the Population Regression Function (PRF) is the conditional mean of Y t given X t . Recall that this is Assumption 3 (A3): For any given value of X, the mean of u is zero: ( | ) 0 t t E u X
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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 7 Another writing form of PRF Instead of writing PRF as 0 1 ( ) t t E Y X we can also equivalently write PRF as 0 1 t t t Y X u where t u is a random variable with mean 0 since ( ) t t t Y E Y u Sample (Estimated) Regression Function, SRF Thus far, we have dealt with the entire population and the PRF. Avoided any consideration of sampling. In most cases, we will never observe the entire population. We have to infer from a sample or samples what the PRF might look like. Note that we are unlikely to know just how close we get to the truth.
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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 8 Each sample we draw can be used to produce a Sample (Estimated) Regression Function (SRF), that is, the estimated regression function: 0 1 ˆ ˆ ˆ t t Y X where ˆ ˆ t t t Y Y u . Hence what we call residuals are as follows: 0 1 ˆ ˆ ˆ ˆ ˆ t t t t t t u Y Y u Y X In other words, the residuals are the differences between the actual and the estimated t Y values. Figure 2 SRF and PRF
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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 9 B. Heteroscedasticity Case Now let us give an example for violation of constant variance assumption. The example data is given below. Table 2 Consumption Expenditures for Different Levels of Disposable Income (Population) Heteroscedasticity Case Disposable Income (X t ) Consumption (Y t ) Average Consump. (Y t ) Disturbances (u t ) ( | ) t t Y E Y X X Cond. Mean of u t ( | ) 0 t t E u X Cond. Variance of u t ( | ) 0 t t t Var u X 1100 500, 520, 540, 600, 850, 1000, 1020, 1050 760 -260, -240, -220, -160, 90, 240, 260, 290 0 [(-260-0) 2 +(-240-0) 2 +(-220-0) 2 + (-160-0) 2 +(90-0) 2 + (240-0) 2 + (260-0) 2 +(290-0) 2 ]/8= 52075 1200 500, 550, 600, 750, 880, 950, 980, 1070, 1100 820 -320, -270, -220, -70, 60, 130, 160, 250, 280 0 [(-320-0) 2 +(-270-0) 2 +(-220-0) 2 + (-70-0) 2 +(60-0) 2 +(130-0) 2 + (160-0) 2 +(250-0) 2 + (280-0) 2 ]/9= 46177.8 1300
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