Lecture 2

Lecture 2 - Moshe Buchinsky Department of Economics UCLA...

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Moshe Buchinsky Economics 103 Department of Economics Introduction to Econometrics UCLA Fall, 2011 Lecture Note 2 The Simple Linear Regression Model I 1T o p i c s t o b e C o v e r e d 1. An Economic Model 2. An Econometric Model 3. Estimating the Regression Parameters 4. Assessing the Least Squares Estimators 5. The Gauss-Markov Theorem 6. The Probability Distributions of the Least Squares Estimators 2A n E c o n o m i c M o d e l As economists we are generally interested in studying relationships between variables For example, economic theory tells us that expenditure on economic goods depends on income Consequently we call | (e.g. expenditure) the “dependent variable” and { (income) the “independent” or “explanatory” variable In econometrics, we recognize that real-world expenditures are random variables ,andw e want to use data to learn about the relationship between { and | . The pdf is a conditional probability density function since it is “conditional” upon an { The conditional mean, or expected value, of | ,cond it iona lon { is H [ | | { ] The expected value of a random variable is called its “mean” value, which is really a contrac- tion of population mean, the center of the probability distribution of the random variable This is not the same as the sample mean, which is simply the arithmetic average of numerical values 1
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Figure 2.1: Probability Distribution of Food Expenditure (i.e. | ), given that income is { =$1 > 000 The conditional variance of | is ± 2 ,wh ichmeasuresthed ispers iono f | about its mean, that is, ² | | { If the parameters ² | | { and ± 2 were known, it would give us some valuable information about the population we are considering Figure 2.1: Probability Distribution of Food Expenditure (i.e. | ), given that income is { > 000 and { =$2 > 000 In order to investigate the relationship between expenditure and income we must build an economic model and then a corresponding econometric model that forms the basis for a quantitative or empirical economic analysis This econometric model is also called a regression model 2
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The simple regression function is written as H [ | | { ]= ± | | { (2.1) = ² 1 + ² 2 {> where ² 1 is the intercept and ² 2 is the slope This model is called a simple regression because there is only one explanatory variable on the right-hand side of the equation Figure 2.2: An Economic Model of LinearRe lat ionsh ipbetweenAverageFood Expenditure and Income The slope of the regression line can be written as: ² 2 = ± H [ | | { ] ± { = gH [ | | { ] g{ > (2.2) where “ ± ”deno te s“changein”and“ gH [ | | { ] @g{ sthede r iva t iveo ftheexpec ted value of | with respect to { 3
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3A n E c o n o m e t r i c M o d e l Figure 2.3: Conditional pdf’s for | at Alternative Levels of Income 3.1 Key Assumptions There are several key assumptions underlying the simple linear regression Assumption 1–Linearity: The mean value of | ,foreachva lueo f { ,isg ivenbythe linear regression H [ | | { ]= ± 1 + ± 2 {> Assumption 2–Constant variance: For each value of { ,theva lueso f | are distributed about their mean values, following pdf’s
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This note was uploaded on 10/01/2011 for the course ECON 103 taught by Professor Sandrablack during the Fall '07 term at UCLA.

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Lecture 2 - Moshe Buchinsky Department of Economics UCLA...

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