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Unformatted text preview: 1 The Capital Asset Pricing Model Wooldridge Chapter 2 & Berndt Chapter 2 1.1 Simple Regression regression analysis e.g. the heights of fathers and their sons e.g. schooling and log wage we have n subjects indexed by i = 1 ;:::;n & two data variables ( x;y ) . a data variable stores a value for each subject: ( x i ;y i ) explain a dependent variable y in some population with an explanatory variable x or determine how a change in x a/ects y . in general, y and x may be related in the population through y i = f ( x i ) + u i where the error term u capture the e/ect of factors in addition to x that determine y . assume f ( & ) is linear: y i = & + & 1 x i + u i . the slope parameter & 1 is the partial derivative of y with respect to x the ceteris paribus e/ect of x on y , with all the other stu/ captured in u held xed. much of econometrics is concerned with one of two potentially di/erent things: the partial derivative of y with respect to x , which is & 1 the derivative of the population conditional mean of y with respect to x : E [ y i j x i ] = E [ & + & 1 x i + u i j x i ] = & + & 1 x i + E [ u i j x i ] , so d dx i E [ y i j x i ] = & 1 + d dx i E [ u i j x i ] . under the zero conditional mean assumption: E [ u i j x i ] = 0 the slope parameter & 1 has an interpretation of partial e/ect: & 1 = d dx i E [ y i j x i ] . conclusion: if u is correlated with x , then we cannot learn the ceteris paribus e/ect of x on y from knowledge of E [ y i j x i ] or d dx i E [ y i j x i ] . we will consider the E [ u i j x i ] 6 = 0 case after midterm exam. estimating & and & 1 recall y i = & + & 1 x i + u i , where ( x i ;y i ) denotes the heights of fathers and their sonsor schooling and log wage. we do not know the population parameters ( & ;& 1 ) so estimate them using a sample from the population. one could plot the data and use the eye ball method to choose the best possible line. let & b & ; b & 1 denote the estimates. then b y i = b & + b & 1 x i is called the tted (or predicted) value of y i . the di/erence b u i y i b y i is called the tted residual for observation i . we can write y i = b & + b & 1 x i + b u i . 1 sample regression function b f ( x i ) = b & + b & 1 x i is called the sample regression function. it is an estimate of the population regression function, f ( x i ) = & + & 1 x i . f ( x i ) is unknown but &xed. b f ( x i ) is obtained for a given sample of data but will vary from sample to sample. 1.2 Return and Risk assume that when investors act in the securities market, their behavior is perfectly rational in the sense that the only concern is assessing returns from their own investments. de&ne the rate of return on an investment as ( p 1 + d & p ) =p , where p 1 = price of security at the end of the time period d = dividends paid during the time period p = price of security at the beginning of the time period the return can be calculated ex post once the investment has been made, but the return is uncertain ex ante before the investment has been made.but the return is uncertain ex ante before the investment has been made....
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- Spring '08