3-16 Deriving the ols estimator Rearranging for B B(1x2-∑x2)=1-∑xy · So overall we have ∑x1y1-Tx∑(x-x)y-j) B ∑ ∑ (3.4) y-所x (35) This method of finding the optimum is known as ordinary least squares OLs)
3-16 Deriving the OLS Estimator • Rearranging for , • So overall we have (3.4) (3.5) • This method of finding the optimum is known as ordinary least squares (OLS). $ − t = − t t (Tx x ) Tyx x y ˆ 2 2 − − − = − − = 2 2 2 ( ) ( )( ) ˆ x x x x y y x Tx x y Txy t t t t t t y x ˆ ˆ = −
What do We Use a and B For 317 In the CaPm example used above, plugging the 5 observations in to make up the formulae given above would lead to the estimates a=-1.74 and b=1. 64. We would write the fitted line as y1=-1.74+1.64x Question: If an analyst tells you that she expects the market to yield a return 20% higher than the risk-free rate next year what would you expect the return on fund xXX to be? Solution: We can say that the expected value ofy="-1.74+ 1.64 X value of x, so plug x=20 into the equation to get the expected value for y: 74+1.64×20=31.06
3-17 What do We Use and For? • In the CAPM example used above, plugging the 5 observations in to make up the formulae given above would lead to the estimates = -1.74 and = 1.64. We would write the fitted line as: • Question: If an analyst tells you that she expects the market to yield a return 20% higher than the risk-free rate next year, what would you expect the return on fund XXX to be? • Solution: We can say that the expected value of y = “-1.74 + 1.64 × value of x”, so plug x = 20 into the equation to get the expected value for y: $ $ $ $ y ˆ i = −1.74 +1.6420 = 31.06 t t y ˆ = −1.74 +1.64x
3-18 accuracy of Intercept estimate Care needs to be exercised when considering the intercept estimate, particularly if there are no or few observations close to the y-axis
3-18 Accuracy of Intercept Estimate • Care needs to be exercised when considering the intercept estimate, particularly if there are no or few observations close to the y-axis: y 0 x
3-19 4 Some terminology The population and the sample The population (s)is the total collection of all objects or people to be studied, for example · Interested in Population of interest predicting outcome the entire electorate of an election a sample is a selection of just some items from the population A random sample is a sample in which each individual item in the population is equally likely to be drawn 随机抽样调査) The size of the sample样本容量:观测/采用个数
3-19 4 Some terminology The Population and the Sample • The population (总体)is the total collection of all objects or people to be studied, for example, • Interested in Population of interest predicting outcome the entire electorate of an election • A sample is a selection of just some items from the population. • A random sample is a sample in which each individual item in the population is equally likely to be drawn. （随机抽样调查） • The size of the sample 样本容量：观测/采用个数
3-20 The prf and the dge The population regression function (PRF)is a description of the model that is thought to be generating the actual data and it represents the true relationship between the variables (i.e. the true values of a and B) The prf is y,=a+x, +u 在本书中,PRF与数据生成过程 data generating process(DGP)是同义词。 在有些书中,两者有所区别。PRF指变量间的真实关 系,而DGP是产生y的实际观测数据的内在过程
3-20 The PRF and the DGP • The population regression function (PRF) is a description of the model that is thought to be generating the actual data ，and it represents the true relationship between the variables (i.e. the true values of and ). • The PRF is • 在本书中 ， PRF 与 数 据 生 成 过 程 data generating process (DGP)是同义词。 • 在有些书中，两者有所区别。PRF指变量间的真实关 系，而DGP是产生 y 的实际观测数据的内在过程。 yt = + xt +ut