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intro-ects-handout-3

# intro-ects-handout-3 - Introductory Econometrics...

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Introductory Econometrics ECON0701 (2009) 19 3. The simple regression model starts with the simplest econometric model: relates a dependent variable ( y ) to a single explanatory variable ( x ) and a constant example: how weekly food expenditure depends on household income questions we are interested, including: If weekly income goes up by \$1000, how much will average weekly food expenditures rise? How much would you predict the expenditure on food to be for a household with an weekly income of \$12,000? 2 methodological questions: (a) Given an economic model involving a relationship between two economic variables, how do we go about specifying the cor- responding econometric model? (Sections 3.1) (b) Given the econometric model and a sample of data on these economic variables, how do we use this information to obtain estimates of the unknown parameters of the relationship con- necting these variables? (Section 3.2) Remaining sections: properties of the ordinary least-sqaures (OLS) estimators 3.1. Developing an econometric model from economic theory: An ex- ample theory: describes the (“average” or “expected”) relationships among economic variables (Figure 3.1: indi ff erence curves, Engel curve)

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Introductory Econometrics ECON0701 (2009) 20 let y = weekly expenditure of a household on food items, and x = weekly household income relationship: y = g ( x ) (3.1) at this level, the relation (3.1) is very general; examples include y = β 0 + β 1 x or y = β 0 x β 1 or ln y = β 0 + β 1 x 2 theoretical prediction: when household income increases, food expenditure goes up as well g 0 y x > 0 (3.2) e.g., y = β 0 + β 1 x where β 1 > 0 3.1.1. Introducing random elements emphasizes the point that in reality, food expenditure of a house- hold with income x is usually di ff erent from that of another household with the same x useful to model food expenditure as a random variable the above theory is reinterpreted as: average weekly household spending on food ( E ( y/x ) ) depends on x , and increases in x
Introductory Econometrics ECON0701 (2009) 21 3.1.2. Linear relationships a further assumption: the relationship between income and con- sumption is linear E ( y | x ) = β 0 + β 1 x (3.1a) β 0 and β 1 : regression parameters; Figure 3.2 slope ( β 1 ): the change in E ( y | x ) for a \$1 change in weekly income; marginal propensity to consume on food β 1 = dE ( y | x ) dx (3.2a) intercept ( β 0 ): the average weekly food expenditure by a house- hold with no (current) income–more comments to come 3.1.3. The simple regression model food expenditure has a probability density function, f ( y ) ; Fig- ure 3.3 if we take a random sample of household with weekly income equal to x = 4800 , the actual expenditure will be scattered around the mean value E ( y | x = 4800) μ y | x =4800 = β 0 + β 1 × 4800 if we were to sample household expenditures at other levels of income (e.g., x = 8000 ), we would expect the sample values to be scattered around their mean value E ( y | x ) = β 0 + β 1 x (Figure 3.4) Introducing the random error ( ε ) term explicitly and assuming a linear relation, the simple regression model is given by y =

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intro-ects-handout-3 - Introductory Econometrics...

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