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University of Guelph
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Economics
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ECON 2740
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David Prescott
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Chapter 3

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Chapter 3Statistical Estimation of The Regression Function31 Statistical EstimationIf the population can be observed there is no statistical problem all the features if the populationare knownThe problem of statistical inference arises when the available information consists of alimited sample that is randomly drawn from the possibly infinitely large population and we want toinfer something about the population using the sample at handStatistical estimation is one aspect of1it concerns the estimation of population parameters such as the population meanstatistical inferenceand variance and the coefficients of a linear regression functionIn Chapter 2 the population regression function of Y given X is defined as the conditional meanfunction of Y given X written as EYXAn important reason for our interest in this functionalrelationship is that it allows us predict Y given values of X and to quantify the effect of a change in X onY measured by say the derivative with respect to XMoreover the conditional predictions of Y that areproduced by the regression function are optimal under specific but generally applicable conditionsThischapter is concerned with the problem of estimating the population regression function using a sampledrawn from the population311 Parametric Versus Nonparametric MethodsFigure 27 and the related discussion illustrates how a sample can be used to estimate apopulation regressionSince the population regression function of Y given X is the conditional mean ofY given X we simply computed a sequence of conditional means using the sample and plotted themNothing in the procedure constrains the shape of the estimated regressionIndeed the empiricalregression of Size given Price the plot of in Figure 27 wanders about quite irregularly althoughas it does so it retains a key feature that we expect of the population regression of S given P namely thatits average slope is steeper than the major axisthe empirical regression starts off below the major axisand then climbs above it The method used to estimate the empirical regression functions in Figure 271 Two other important inference problems are hypothesis testing and the prediction of randomvariablesEconometrics Text by D M Prescott Chapter 3 2can be described as nonparametricWhile there is a huge literature on nonparametric estimation thisbook is concerned almost entirely with parametric modelsTo illustrate the distinction between parametric and nonparametric methods consider theequation YabXThis equation has two parameters or coefficients a and b and clearly therelationship between Y and X is linearBy varying the values of a and b the lines height and slope canbe changed but the fundamental relationship is constrained to be linearIf a quadratic term and one2the relationship between Y and X becomes more flexiblemore parameter is added YabXcXthan the linear functionIndeed the quadratic form embraces the linear form as a special case set c0 But the linear form does not embrace the quadratic formno values of a and b can make the linearequation quadraticOf course the three parameter quadratic equation is also constrainedA quadraticfunction can have a single maximum or a single minimum but not bothQuadratic functions are also2symmetric about some axisIf further powers of X are added each with its own parameter therelationship becomes increasingly flexible in terms of the shape it can takeBut as long as the number ofparameters remains finite the shape remains constrained to some degree The nonparametric case isparadoxically not the one with zero parameters but the limiting case as the number of parametersincreases without boundAs the number of terms in the polynomial tends to infinity the functionalrelationship becomes unconstrainedit can take any shapeAs noted above the method used toconstruct the empirical regressions in Figure 27 did not constrain the shape to be linear quadratic or anyother specific functional relationshipIn that sense the method used in Chapter 2 to estimate thepopulation regression can be called nonparametricIn the context of regression estimation the great appeal of nonparametric methods is that they do not impose a predetermined shape on the regression functionwhich seems like a good idea in theabsence of any information as to the shape of the population regressionHowever there is a cost3 the nonparametricassociated with this flexibility and that concerns the sample sizeTo perform wellestimator generally requires a large sample the empirical regressions in Figure 27 used a sample ofalmost 5000 observationsIn contrast parametric methods that estimate a limited number of parameterscan be applied when samples are relatively smallThe following examples bypass the statistical aspect22 The graph of YabXcX is symmetric about the line X b2c3 The meaning of performing well will be discussed later in the chapter Econometrics Text by D M Prescott Chapter 3 3of the argument but nevertheless provide some intuitionIf you know that Y is a linear function of Xthen two points 2 observationsare sufficient to locate the line and to determine the two parameters If you know the relationship is quadratic just three points are sufficient to plot the unique quadraticfunction that connects the three points and therefore three observations will identify the three parametersof the quadratic equationThe relationship continues in general n points will determine the n parametersth order polynomialof an n32 Principles of EstimationAs discussed in Chapter 2 there are examples of bivariate distributions in which the populationregression functions are known to be linearIn the remainder of this chapter we will be concerned with linear population regressions and the methods that can be used to estimate themWe begin with adiscussion of alternative approaches to statistical estimationall of which are parametric321 The Method of MomentsThe quantitiesare referred to as the first second and third uncentred moments of the random variable XThe centredmoments are measured around the meanThe Method of Moments approach to estimating these quantities is to simply calculate their sampleequivalents all of which take the form of averagesTable 31 provides the details for the first twomomentsNotice the parallels between the expressions for the population moments and their samplecounterpartsFirst the estimator uses instead of the expectation operator EBoth take anaverage one in the sample the other in the population Second the estimator is a function of the whereas the population moment is defined in terms of the random variable Xobservations Xi

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