口 Chapter5: Basic Ideas of linear regression the two- variable model
Chapter 5: Basic Ideas of Linear Regression: the Two-Variable Model
The Meaning of Regression The Sample Regression Analysis Function (SRF) The Population Regression The le Special Meaning Function (PRF) of the Term " Linear” Regression Stochastic Specification of the TTwo-Variable versus Population Regression Function(PRF) Multiple Linear Regression The Nature of the Stochastic Estimation of Error Term Parameters
The Population Regression Function(PRF) The Meaning of Regression Analysis Stochastic Specification of the Population Regression Function(PRF) The Nature of the Stochastic Error Term The Sample Regression Function（SRF） The Special Meaning of the Term “Linear” Regression Two –Variable versus Multiple Linear Regression Estimation of Parameters
5.1 The Meaning of Regression Analysis 1. Regression analysis-the study of the relationship between one variable Y(the explained, or dependent variable)and one or more other variables Xxs (explanatory, or independent variables) Y: the explained, or dependent variable X/XS: the explanatory, or independent variables
5.1 The Meaning of Regression Analysis 1.Regression analysis——the study of the relationship between one variable Y (the explained，or dependent variable) and one or more other variables X/Xs (explanatory，or independent variables)． Y: the explained，or dependent variable X/Xs: the explanatory，or independent variables
5.1 The Meaning of Regression Analysis Ms 2.The objective of regression analysis (1 To estimate the mean, or average, value of the dependent variable, given the values of the independent variables (2) To test hypotheses about the nature of the dependence-hypotheses suggested by the underlying economic theory (3) To predict, or forecast, the mean value of the dependent variable, given the values of the independent variable(s)
5.1 The Meaning of Regression Analysis 2.The objective of regression analysis: （1）To estimate the mean，or average， value of the dependent variable，given the values of the independent variables． （2）To test hypotheses about the nature of the dependence-hypotheses suggested by the underlying economic theory． （3）To predict，or forecast，the mean value of the dependent variable，given the values of the independent variable(s)．
5.1 The Meaning of Regression Analysis 03. Note: (1) Regression does not necessarily imply causation. Causality must be justified, or inferred, from the theory that underlies the phenomenon that is tested empiricall (2) (3)
5.1 The Meaning of Regression Analysis 3. Note: (1) Regression does not necessarily imply causation．Causality must be justified， or inferred，from the theory that underlies the phenomenon that is tested empirically． （2） （3）
5.2 The Population Regression Function(PRF) a1. Population Regression Line(PRL) 0 PRI a line that tells us how the average/mean value of y (or any dependent variable)is related to each value of X(or any independent variable). a line that passes through the conditional means ofY D E(YXI)=B1+B2Xi (5.1) E(YXI: the mean, orexpected, value of Y corresponding to, or conditional upon, a given value of X. conditional expectation or conditional expected value ofy BI and B2: the parameters, the regression coefficients Bl: the intercept(coefficient) B2: the slope(coefficient) The slope coefficient measures the rate of change in the(conditional) mean value of Y per unit change inⅩ
5.2 The Population Regression Function(PRF) 1.Population Regression Line (PRL) PRL——a line that tells us how the average/mean value of Y (or any dependent variable) is related to each value of X(or any independent variable)． ——a line that passes through the conditional means of Y． E(Y|Xi) = B1+B2Xi (5.1) E(Y|Xi):the mean，or expected，value of Y corresponding to，or conditional upon，a given value of X． ～conditional expectation or conditional expected value of Y． B1 and B2: the parameters, the regression coefficients． B1: the intercept (coefficient) B2: the slope (coefficient)．—— The slope coefficient measures the rate of change in the (conditional) mean value of Y per unit change in X．
5.2 The Population Regression Function(PRF) a Population Regression Function (PRF) E(Xi)=B+B2Xi m Note: Usually expressions like E(YXi is simply written as E(Y), with the explicit understanding that the latter in fact stands for the former
2. Population Regression Function（PRF） E(Y|Xi) = B1+B2Xi 3. Note: Usually expressions like E(Y|Xi) is simply written as E(Y)，with the explicit understanding that the latter in fact stands for the former． 5.2 The Population Regression Function(PRF)
5.3 Stochastic Specification of the Population Regression Function(PRF n The deterministic/nonstochastic PRF: E(YXI)=B1+B2Xi (51) which represents the means of the various Y values corresponding to the specified Xs n The stochastic prF: Yi-B1+ B2Xi+ui (5.2) which tells us how individual Ys vary around their mean values due to the presence of the stochastic error term ()(B1+B2XI), the systematic, or deterministic, component of PrF (2)ui, which may be called the nonsystematic, or random component of PRF where u is known as the stochastic, or random error, term, or simply the error term----a random variable (rv)
5.3 Stochastic Specification of the Population Regression Function(PRF) The deterministic/nonstochastic PRF: E(Y|Xi)=B1+B2Xi…… (5.1) which represents the means of the various Y values corresponding to the specified Xs， The stochastic PRF: Yi=B1+B2Xi+μi (5.2) which tells us how individual Ys vary around their mean values due to the presence of the stochastic error term (1)(B1+B2Xi)，the systematic, or deterministic, component of PRF (2)μi，which may be called the nonsystematic, or random component of PRF． where μ is known as the stochastic, or random error, term, or simply the error term----a random variable（r.v.）
5.4 The Nature of the stochastic Error Term @the influence of those variables that are not explicitly included in the model ⑧some“ intrinsic” randomness. such as human behavior, that cannot be explained in the model. ⑧ errors of measurement Osome variables might affect Y, but their combined influence on Y is so small and nonsystematic Note: Error term plays an extremely crucial role In regression model
5.4 The Nature of the Stochastic Error Term. the influence of those variables that are not explicitly included in the model． Some “intrinsic” randomness, such as human behavior, that cannot be explained in the model. errors of measurement some variables might affect Y, but their combined influence on Y is so small and nonsystematic. Note: Error term plays an extremely crucial role in regression model
5.5 The Sample Regression Function (SRF) 1. Sample Regression Lines(SRLS) and Sample regression Function (SRF) o Deterministic SRF: Yi= b1+ b2 Xi (5.3) 无法显示该图片 o Yi=the estimator of E(YXi, the estimator of the population conditional mean n bl=the estimator of B1 n b2=the estimator of B2 n Stochastic srf: Y1=b1+b2Xi+ei (5.4) n ei: the residual term, or simply the residual, it is analogous to ui and can be regarded as the where ei is the estimator of ui
5.5 The Sample Regression Function （SRF） Deterministic SRF: (5.3) =the estimator of E(Y|Xi)，the estimator of the population conditional mean b1=the estimator of B1 b2=the estimator of B2 Stochastic SRF: Y1=b1+b2Xi+ei (5.4) ei: the residual term，or simply the residual, it is analogous to μi and can be regarded as the where ei is the estimator of μi． 1.Sample Regression Lines (SRLs) and Sample Regression Function（SRF）． Yi = b1 + b2Xi ˆ Yi ˆ