This is an initial assumption of our estimation we

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This is an initial assumption of our estimation. We could narrow this functional form to: 0 1 ( | ) t t E Y X X X or simply 0 1 ( ) t E Y X This is the (linear) Population Regression Function (PRF) 1 : 0 1 t X . Notice that, here the PRF is the straight line connecting these conditional means. Clearly, consumption expenditures “on average” increase with disposable income. 1 Or, True Regression Function.
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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 4 Geometrically, the Population Regression Function (PRF) is simply the locus of average consumption expenditures of our population at different income level s 2 . Analytically the equation of PRF 3 can be found using any two data points. For example, ( | 1100) 760 E Y X , 1100 X and ( | 1200) 820 E Y X , 1200 X can be used: 0 1 1100 760 0 1 1200 820 Solving this system yields 0 100 and 1 0.6 . Hence the equation of PRF is as follows: ( | ) 100 0.6 t t E Y X X X Role of Disturbance The PRF tells us the “average” consumption expenditures for a given level of household income. But we know that any “particular” household is unlikely to be on this function since there are stochastic deviations from these average consumption expenditure levels at different levels of income. The “average” consumption expenditures at different income levels, in terms of PRF , can be expressed as ( Data Generating Process, DGP ): ( | ) t t t Y E Y X X u where t u is a random variable with mean 0 (disturbance). 2 Or, “ the conditional means or expectations of the dependent variable for fixed values of the explanatory variable(s) . 3 0 1 ( | ) t t E Y X X 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 5 Economic relationships (other than accounting identities) are not exact and t u plays an important role in econometrics. Figure 2 Probability Mass Functions 4 for Y at Different Levels of Income (Homoscedasticity Case) The figure shows the conditional probability of consumptions with family incomes of 1100, 1200, 1300 and 1400 TL. The mean of the conditional distribution of consumptions, given the family income level, E(Y|X), is the population regression line 0 1 X . At a given value of X, Y is distributed around the regression line and the disturbance, 0 1 ( ) u Y X , has a conditional mean of zero for all values of X. Said differently, the distribution of t u , conditional on t X X , has a mean of zero; stated mathematically, ( | ) 0 t t E u X X , or, in simply ( | ) 0 t t E u X . 4 Since the distribution in our example is discrete, the distribution function it is called probability mass function .
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