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Unformatted text preview: Adapted from Vera Tabakova’s notes ECON 4550 Econometrics Memorial University of Newfoundland 5.1 Model Specification and Data 5.2 Estimating the Parameters of the Multiple Regression Model 5.3 Sampling Properties of the Least Squares Estimator 5.4 Interval Estimation 5.5 Hypothesis Testing for a Single Coefficient 5.6 Measuring GoodnessofFit β 2 = the change in monthly sales S ($1000) when the price index P is increased by one unit ($1), and advertising expenditure A is held constant = β 3 = the change in monthly sales S ($1000) when advertising expenditure A is increased by one unit ($1000), and the price index P is held constant = 1 2 3 S P A = β + β + β ( held constant) A S S P P ∆ ∂ = ∆ ∂ ( held constant) P S S A A ∆ ∂ = ∆ ∂ Figure 5 .1 The multiple regression plane Slide 54 Slide 55 The introduction of the error term, and assumptions about its probability distribution, turn the economic model into the econometric model in (5.2). 1 2 3 ( ) i i i i i i S E S e P A e = + = β + β + β + 1 2 2 3 3 i i i K iK i y x x x e = β + β + β + + β + L ( 29 ( 29 other 's held constant k k k x E y E y x x ∆ ∂ β = = ∆ ∂ 1 2 2 3 3 i i i y x x e = β + β + β + 1. Each random error has a probability distribution with zero mean. Some errors will be positive, some will be negative; over a large number of observations they will average out to zero. ( ) i E e = 1. Each random error has a probability distribution with variance σ 2 . The variance σ 2 is an unknown parameter and it measures the uncertainty in the statistical model. It is the same for each observation, so that for no observations will the model uncertainty be more, or less, nor is it directly related to any economic variable. Errors with this property are said to be homoskedastic . 2 var( ) i e = σ 1. The covariance between the two random errors corresponding to any two different observations is zero. The size of an error for one observation has no bearing on the likely size of an error for another observation. Thus, any pair of errors is uncorrelated. cov( , ) i j e e = 1. We will sometimes further assume that the random errors have normal probability distributions. ( 29 2 ~ 0, i e N σ The statistical properties of y i follow from the properties of e i . 1. The expected (average) value of y i depends on the values of the explanatory variables and the unknown parameters. It is equivalent to . This assumption says that the average value of y i changes for each observation and is given by the regression function . 1 2 2 3 3 ( ) i i i E y x x = β + β + β ( ) i E e = 1 2 2 3 3 ( ) i i i E y x x = β + β + β 1....
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This note was uploaded on 01/26/2011 for the course STAT 2550 taught by Professor Wang during the Spring '10 term at Memorial University.
 Spring '10
 wang
 Least Squares

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