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nullnullnull Figure 2.1a Probability distribution of food expenditure y given income x = $1000Slide 2-*null Figure 2.1b Probability distributions of food expenditures y given incomes x = $1000 and x = $2000 null The simple regression function null Figure 2.2 The economic model: a linear relationship between average per person food expenditure and incomenull Slope of regression line “Δ” denotes “change in” null Figure 2.3 The probability density function for y at two levels of incomeSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullnull2.2.1 Introducing the Error Term The random error term is defined as Rearranging gives y is dependent variable; x is independent variable Slide 2-*null The expected value of the error term, given x, is The mean value of the error term, given x, is zero. Slide 2-*null Figure 2.4 Probability density functions for e and ySlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*null Figure 2.5 The relationship among y, e and the true regression lineSlide 2-*null Slide 2-*null Figure 2.6 Data for food expenditure example Slide 2-*null2.3.1 The Least Squares Principle The fitted regression line is The least squares residual Slide 2-*null Figure 2.7 The relationship among y, ê and the fitted regression line Slide 2-*nullAny other fitted line Least squares line has smaller sum of squared residuals If and then SSE < SSE* Slide 2-*null Least squares estimates for the unknown parameters β1 and β2 are obtained by minimizing the sum of squares function Slide 2-*nullThe Least Squares Estimators Slide 2-*null2.3.2 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: Slide 2-*null Figure 2.8 The fitted regression line Slide 2-*null2.3.3 Interpreting 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. Slide 2-*null2.3.3a Elasticities 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 Slide 2-*nullThe 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. Slide 2-*null2.3.3b Prediction 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. Slide 2-*null2.3.3c Examining Computer Output Figure 2.9 EViews Regression Output Slide 2-*null2.3.4 Other Economic Models The “log-log” model Slide 2-*null2.4.1 The estimator b2 Slide 2-*null2.4.2 The 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 Slide 2-*null2.4.3 Repeated Sampling Slide 2-*nullSlide 2-*The variance of b2 is defined as Figure 2.10 Two possible probability density functions for b2 null2.4.4 The 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: Slide 2-*null2.4.4 The 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 . Slide 2-*nullSlide 2-*The 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 precision null Slide 2-*nullThe 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. Slide 2-*nullThe 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. Slide 2-*nullIf we make the normality assumption (assumption SR6 about the error term) then the least squares estimators are normally distributed Slide 2-*nullThe variance of the random error ei is if 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 are Slide 2-*nullThe 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 is Slide 2-*nullSlide 2-*Replace the unknown error variance in (2.14)-(2.16) by to obtain: nullSlide 2-*The square roots of the estimated variances are the “standard errors” of b1 and b2. null Slide 2-*nullSlide 2-*The 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. nullSlide 2-*For the food expenditure data the estimated covariance matrix is: nullSlide 2-* nullSlide 2-*nullSlide 2-*nullSlide 2-*null Figure 2A.1 The sum of squares function and the minimizing values b1 and b2 Slide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*nullSlide 2-*Let be any other linear estimator of β2. Suppose that ki = wi + ci.nullSlide 2-*nullSlide 2-*