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ECON 6002 Econometrics Memorial University of Newfoundland. The Simple Linear Regression Model: Specification and Estimation. Chapter 2. Adapted from Vera Tabakova’s notes. Chapter 2: The Simple Regression Model. SECOND. Estimating the Regression Parameters

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ECON 6002 Econometrics Memorial University of NewfoundlandThe Simple Linear Regression Model: Specification and EstimationChapter2Adapted from Vera Tabakova’s notes Chapter 2: The Simple Regression ModelSECONDEstimating the Regression Parameters Assessing the Least Squares Estimators The Gauss-Markov Theorem The Probability Distributions of the Least Squares Estimators Estimating the Variance of the Error Term Principles of Econometrics, 3rd Edition2.2 An Econometric ModelFigure 2.5 The relationship among y, e and the true regression linePrinciples of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersPrinciples of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersFigure 2.6 Data for food expenditure exampletwoway (scatter food_exp income)Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersThe Least Squares Principle The fitted regression line is The least squares residual Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersFigure 2.7 The relationship among y, ê and the fitted regression linePrinciples of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersAny other fitted line Least squares line has smaller sum of squared residuals Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersLeast squares estimates for the unknown parameters β1 and β2are obtained by minimizing the sum of squares function Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersThe Least Squares Estimators Principles of Econometrics, 3rd Edition2.3 Estimating The Regression Parameters2.3.2 Estimates for the Food Expenditure Function A convenient way to report the values for b1 and b2 is to write out the estimated or fitted regression line: Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersFigure 2.8 The fitted regression linePrinciples of Econometrics, 3rd Edition2.3 Estimating The Regression Parameterstwoway (scatter food_exp income) (lfit food_exp income)Principles of Econometrics, 3rd EditionFigure 2.8 The fitted regression lineSlide 2-142.3 Estimating The Regression ParametersInterpreting the Estimates The value b2 = 10.21 is an estimate of 2, the amount by which weekly expenditure on food per household increases when household weekly income increases by $100. Thus, we estimate that if income goes up by $100, expected weekly expenditure on food will increase by approximately $10.21. Strictly speaking, the intercept estimate b1 = 83.42 is an estimate of the weekly food expenditure on food for a household with zero income. Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersElasticities Income elasticity is a useful way to characterize the responsiveness of consumer expenditure to changes in income. The elasticity of a variable y with respect to another variable x is In the linear economic model given by (2.1) we have shown that Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersThe elasticity of mean expenditure with respect to income is A frequently used alternative is to calculate the elasticity at the “point of the means” because it is a representative point on the regression line. Principles of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersPrinciples of Econometrics, 3rd EditionA frequently used alternative is to calculate the elasticity at the “point of the means” because it is a representative point on the regression line. In Stata: mfx compute, eyex at(mean) mfx compute, eyex at(income=10) Slide 2-182.3 Estimating The Regression ParametersPrediction Suppose that we wanted to predict weekly food expenditure for a household with a weekly income of $2000. This prediction is carried out by substituting x = 20 into our estimated equation to obtain We predict that a household with a weekly income of $2000 will spend $287.61 per week on food. Principles of Econometrics, 3rd Edition2.3 Estimating The Regression Parameters Examining Computer Output Figure 2.9 EViews Regression OutputPrinciples of Econometrics, 3rd Edition2.3 Estimating The Regression ParametersOther Economic Models The “log-log” model Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe estimator b2 Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe Expected Values of b1 and b2 We will show that if our model assumptions hold, then , which means that the estimator is unbiased. We can find the expected value of b2 using the fact that the expected value of a sum is the sum of expected values using and Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsRepeated SamplingPrinciples of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe variance of b2 is defined as Figure 2.10 Two possible probability density functions for b2Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe Variances and Covariances of b1 and b2 If the regression model assumptions SR1-SR5 are correct(assumption SR6 is not required),then the variances and covariance of b1 and b2 are: Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe Variances and Covariances of b1 and b2 The larger the variance term , the greater the uncertainty there is in the statistical model, and the larger the variances and covariance of the least squares estimators. The larger the sum of squares, , the smaller the variances of the least squares estimators and the more precisely we can estimate the unknown parameters. The larger the sample size N, the smaller the variances and covariance of the least squares estimators. The larger this term is, the larger the variance of the least squares estimator b1. The absolute magnitude of the covariance increases the larger in magnitude is the sample mean , and the covariance has a sign opposite to that of . Principles of Econometrics, 3rd Edition2.4 Assessing the Least Squares EstimatorsThe variance of b2 is defined as Figure 2.11 The influence of variation in the explanatory variable x on precision of estimation (a) Low x variation, low precision (b) High x variation, high precisionPrinciples of Econometrics, 3rd Edition2.5 The Gauss-Markov TheoremPrinciples of Econometrics, 3rd Edition2.5 The Gauss-Markov TheoremThe estimators b1 and b2 are “best” when compared to similar estimators, those which are linear and unbiased. The Theorem does not say that b1 and b2 are the best of all possible estimators. The estimators b1 and b2 are best within their class because they have the minimum variance. When comparing two linear and unbiased estimators, we always want to use the one with the smaller variance, since that estimation rule gives us the higher probability of obtaining an estimate that is close to the true parameter value. In order for the Gauss-Markov Theorem to hold, assumptions SR1-SR5 must be true. If any of these assumptions are not true, then b1 and b2 are not the best linear unbiased estimators of β1 and β2. Principles of Econometrics, 3rd Edition2.5 The Gauss-Markov TheoremThe Gauss-Markov Theorem does not depend on the assumption of normality (assumption SR6). In the simple linear regression model, if we want to use a linear and unbiased estimator, then we have to do no more searching. The estimators b1 and b2 are the ones to use. This explains why we are studying these estimators and why they are so widely used in research, not only in economics but in all social and physical sciences as well. The Gauss-Markov theorem applies to the least squares estimators. It does not apply to the least squares estimates from a single sample. Principles of Econometrics, 3rd Edition2.6 The Probability Distributions of the Least Squares EstimatorsIf we make the normality assumption (assumption SR6 about the error term) then the least squares estimators are normally distributed Principles of Econometrics, 3rd Edition2.7 Estimating the Variance of the Error TermThe variance of the random error eiisif the assumption E(ei)= 0 is correct.Since the “expectation” is an average value we might consider estimating σ2 as the average of the squared errors,Recall that the random errors arePrinciples of Econometrics, 3rd Edition2.7 Estimating the Variance of the Error TermThe least squares residuals are obtained by replacing the unknown parameters by their least squares estimates,There is a simple modification that produces an unbiased estimator, and that isPrinciples of Econometrics, 3rd Edition2.7.1 Estimating the Variances and Covariances of the Least Squares EstimatorsReplace the unknown error variance in (2.14)-(2.16) by to obtain: Principles of Econometrics, 3rd Edition2.7.1 Estimating the Variances and Covariances of the Least Squares EstimatorsThe square roots of the estimated variances are the “standard errors” of b1 and b2. Principles of Econometrics, 3rd Edition2.7.2 Calculations for the Food Expenditure DataPrinciples of Econometrics, 3rd Edition2.7.2 Calculations for the Food Expenditure DataThe estimated variances and covariances for a regression are arrayed in a rectangular array, or matrix, with variances on the diagonal and covariances in the “off-diagonal” positions. Principles of Econometrics, 3rd Edition2.7.2 Calculations for the Food Expenditure DataFor the food expenditure data the estimated covariance matrix is: In STATA:estat vcePrinciples of Econometrics, 3rd Edition2.7.2 Calculations for the Food Expenditure DataPrinciples of Econometrics, 3rd EditionKeywordsassumptions asymptotic B.L.U.E. biased estimator degrees of freedom dependent variable deviation from the mean form econometric model economic model elasticity Gauss-Markov Theorem heteroskedastic homoskedastic independent variable least squares estimates least squares estimators least squares principle least squares residuals linear estimator prediction random error term regression model regression parameters repeated sampling sampling precision sampling properties scatter diagram simple linear regression function specification error unbiased estimator Principles of Econometrics, 3rd Edition

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