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by: Kobe Dare


Kobe Dare
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Class Notes
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This 10 page Class Notes was uploaded by Kobe Dare on Thursday October 22, 2015. The Class Notes belongs to THTR 21 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/226861/thtr-21-university-of-california-santa-barbara in Theatre at University of California Santa Barbara.




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Date Created: 10/22/15
Oct 21 2008 LEC 8 ECON 140A240Al L Phillips Correlation and Analysis of Variance I Introduction Pursuing the results from the ordinary least squares estimates of the linear model from the previous lecture we investigate correlation measures of goodness of t and analysis of variance Then we turn to issues of central tendency and dispersion for the parameter estimates of the intercept and slope ll Correlation Recall that in Lectures Four and Six we introduced the covariance between y and x as a measure of the relationship between these variables Ey 7 Eyx 7 Ex Covyx 1 The covariance depends on units of measurement so it is not a relative measure We can go back to the linear model Eq 4 from the previous lecture and see the relationship between the variance of y and the covariance of y and x yiabxiui 2 Taking expectations EyabExEu 3 The mean of the error term E u is zero by assumption and we have seen that the sample mean ofthe estimated error is zero ie 2 i n 0 1 Subtract Eq 3 from Eq 2 to write in deviation form yEybx7Exu 4 Multiply both sides by x 7 Ex and take expectations If the residual u is independent of the explanatory variable which is another assumption of regression analysis then Ey Eyx 7 Ex b Ex 7 Ex2 E x 7 Ex u b Var x 5 Oct 21 2008 LEC 8 ECON 140A240A2 L Phillips Correlation and Analysis of Variance Solving for the slope b b Cov yXVar X 6 We can estimate the parameter b using the method of moments which substitutes the sample estimates for the population entities of Cov yX and Var X I Z yi Z yiIIXXi Z XinXi Z XinXi39 Z Xin 7 This is the same formula as the ordinary least squares solution given by Eq 15 in the previous lecture To see this rewrite Eq 7 above as 132 yi39 Wm fc Z xv if 8 and expanding 13 2 y xi y xi fyi 7 Z xi2 2 ix x 29 and taking summations ISZ yi m y 2 m fez yi n7x HZ x02 2 fez Xi mow K10 and collecting terms ISZ ininch Z mlnoch 11 and multiplying top and bottom by nn 13 n2 yi KHZ yi Z Xi n2 MHZ Xi 2 12 the same as Eq 15 as promised We can use Eq 4 to pursue the relationship between variances of y X and u Square both sides of Eq 4 and take eXpectations Oct 21 2008 LEC 8 ECON 140A240A3 L Phillips Correlation and Analysis of Variance Ey 7 By2 7 b2 EX 7 Ex2 2b EX 7 EXu E u2 13 or VaryEszarXVar u 14 since another assumption of least squares as discussed above is that the explanatory variable X is independent of the error u so their covariance is zero ie EX 7 EXu E 0 Thus the total variance in y can be decomposed into two parts the variance explained by y s dependence on X b2 Var X called the signal and the uneXplained variance Var u called the noise Combining Eq s 6 and 14 Var y E Cov yX2 Var X Var u 15 And dividing by the variance of y l E Cov yX2 Var XVar y Var uVar y 16 where l 7 Var uVar y is one minus the ratio of the uneXplained variance to the total variance ie l 7 Var uVar y E l 7 uneXplained variancetotal variance 17 E total variance 7 uneXplained variance total variance 18 E eXp1ained variancetotal variance 19 and from Eq 16 this fraction of the total variance that is eXp1ained is eXp1ained variancetotal variance E Cov yX2 Var XVar y 20 Note the covariance squared divided by the variance of y and the variance of X cancels out the units of measurement leaving a relative measure called R2 the coef cient of determination This coef cient which measures the fraction of the variance in y J 39 39 J by J r J on X is quoty a measure of goodness of t 1 Oct 21 2008 LEC 8 ECON 140A240A4 L Phillips Correlation and Analysis of Variance In bivariate regression of y on X the coefficient of determination R2 is just the square of the correlation coefficient r between y and X which is a relative unitless measure of the interdependence of y and X VRZ Cov ym Vary Varx r 21 The sample correlation coefficient f can be estimated by the method of moments substituting sample estimates of sums of squares for the covariance and variances VA Zyi J7lxi C n 1 vyi J7 2 xquot 1 xi f 2 xn 1 or VA Zyi J7lxi 761V Vyi J7 2 xi 7 2 l 22 The correlation coefficient ranges between minus one if the correlation is negative and perfect through zero for no correlation and up to one if the correlation is positive and perfect ie 1S r 1 In multivariate regression where y depends on two or more eXplanatory variables or regressors the estimated coefficient of determination I 2 can be calculated from the sum of squared residuals and the sum of squared deviations of y around its mean 2172 izZ yi 72 23 111 Analysis of Variance The results of a regression analysis can be summarized in a table of analysis of variance or ANOVA as depicted in Table 1 Oct 21 2008 LEC 8 ECON 140A240A5 L Phillips Correlation and Analysis of Variance Table 1 Table of Analysis of Variance Bivariate Regression Source of Variation Sum of Squares Degrees of Freedom Mean Square Explainedbyx 522 Xi ff 1 522 xi f21 Unexplained 2L 2 112 Zr 2 n2 Total 2 yi y 2 n1 2 yi y 2 n1 A key to understanding ANOVA is that the total sum of squared deviations of the dependent variable y from its mean can be partitioned into the explained sum of squares and the unexplained sum of squares in a fashion parallel to how we partitioned the population variance To see this combine Eq 6 L20 yz39 t with Eq 10 J70 51 5 both from the previous chapter to obtain Wyi 5I bxil 5I b 24 iyi J7 bxi f 25 Squaring the observed residual and summing 2mm 2 Zam y 2 b 2 xi 1272 bxz39 anym y 26 Note from Eq 8 that bZxi 7 2 Zyi xi 7 27 and substituting the left hand side of Eq 27 for the right hand side as it appears in Eq 26 we obtain Oct 21 2008 LEC 8 ECON 140A240A6 L Phillips Correlation and Analysis of Variance Emmi ZyiJ72 13 2xi 12 28 ie the residual sum is the difference between the total sum of squares for y and that explained by the dependence of y on x which is depicted in the second column of Table 1 One degree of freedom is lost in a sample size ofn in calculating the sample mean which is used to calculate the total sum of squares for the dependent variable Two degrees of freedom are lost in estimating the two regression parameters 51 and 5 necessary to calculate f and the sum of observed squared residuals for a sample of size 11 That leaves one degree of freedom for the observed mean square as indicated in the third column of Table l The mean squares are just the sums of squares divided by their respective degrees of freedom ie sums in column two divided by degrees of freedom in column three and are listed in column 4 The ratio of the explained mean square to the unexplained mean square has the F distribution with l and n2 degrees of freedom 1 in the numerator and n2 in the denominator sz 132 2 xi 7c2l Edim2 29 This F test can be used to determine whether x significantly explains the variance in y or equivalently whether the goodness of fit is statistically signi cant at some critical level at say 5 This F test can also be conducted with the coefficient of determination Recall that R2 is the ratio of explained variance to total variance or in terms of the estimated Oct 21 2008 LEC 8 ECON 140A240A7 L Phillips Correlation and Analysis of Variance coefficient of determination I 2 the ratio of the explained sum of squares to the total sum of squares As expressed in Eq 23 1 I 2 is the ratio of the unexplained sum of squares to the total sum of squares Thus the ratio of I 21 I 2 is the ratio of the explained sum of squares to the unexplained sum of squares and multiplying by n 72 1 we have the F statistic F1n2 1 211 1 2n 72 n 7 211 21 1 21 30 As an example of a Table of ANOVA we use our data set from Table 1 in lecture seven and the estimation of the capital asset pricing model as illustrated in Figure 4 of that lecture Table 2 Table of Analysis of Variance Bivariate Regression CAPM Mean Square 4063589 40636 1700485 154590 The sum of squared residuals from the regression is 4063589 and divided by ten the unexplained mean square is 40636 The stande deviation in the dependent variable is 3931788 and squaring and multiplying by eleven the total sum of squares is 1700485 The explained sum of squares can be calculated by difference as 1294126 The F statistic with one degree of freedom in the numerator and ten degrees of freedom in the denominator is 31486 The critical value of F1 10 for a level of significance ofoc 5 is 496 so we can reject the null hypothesis that the market has no explanatory power Oct 21 2008 LEC 8 ECON 140A240A8 L Phillips Correlation and Analysis of Variance for the UC stock index fund The coefficient of determination is 0761 and gives an equivalent F statistic Three fourths of the variation in the UC stock index fund is explained by variation in the market with the remainder of the volatility being speci c to the UC stock index fund The square root of the unexplained mean square 20158 is an estimate of the square root of the variance of the error u for the regression 639 u and is called the standard error of the regression IV The Mean and Variance of the OLS Slope Estimate It is useful to express the linear model in deviation form For the sake of notation start with Eq 2 yi a b xi ui 31 and sum over I Z yi na b 2 xi Z ui 32 and divide by n 7abf 33 Subtract Eq 33 from Eq 31 to obtain the linear model in deviation form Yi 7 bXi J C 11i 17 34 Use the right hand side of Eq 34 to substitute for yi 7 in the expression for the estimate of the slope given by Eq 8 3 Z bXi i ui 17Xi 761Z xi Hz 35 5bZ ui l7Xi J 6 Z xi Hz 36 Oct 21 2008 LEC 8 ECON 140A240A9 L Phillips Correlation and Analysis of Variance Take expectations ofboth sides of Eq 36 1313b Z Xi J 6 Eui z Xi Hz 37 Note that E ui 0 for all observations i and E L7 ln ZEui0 as well so the expected value ofthe estimated slope b is the model slope parameter b and as a consequence we say the estimate is unbiased Use Eq 36 to subtract b from 5 and separate the summation in the numerator into two terms 5 39b 1Z Xi39 Hz 2 ui Xi39 J7C 39 17 Z Xi39 J75 Since 2 xi Tc 0 we obtain 3 b 1Z Xi Hz 2 ui Xi J 38 and squaring both sides and taking expectations we obtain the variance of the estimated slope parameter EU b2 1 2 xi J 122 x1 9 2E u12 x2 EYE u22 x1 7cxz 7cE u1u2 39 An assumption of OLS is that the error terms are independent so Euiui 0 all i and j and that the variance of the error is homoskedastic ie the same for all observations so Eu12 Eu22 etc so that Eq 38 simpli es to BUS b212 xi c2zcz2 xigt c12czZ xi if 40 z Smce 6 1s unknown we use the unexplamed mean square as an est1mate Oct 21 2008 LEC 8 ECON 140A240A10 L Phillips Correlation and Analysis of Variance The estimate of the variance for the estimated slope b for the CAPM is VaruS EU b2 lt9 zZxi 7c 2 406362528868 0016068 41 And taking the square root the estimated standard deviation for 5 is 01268 V Hypothesis Tests About the Slope Using the estimated CAPM as an example the null hypothesis that the UC stock index fund does not depend on the market can be tested by the conjecture that the slope is zero ie H0 b 0 Versus the alternative hypothesis that the slope is not zero Ha b at 0 Forming a tstatistic with 10 degrees of freedom t 13 b 613 0715 7 001268 563 and for 10 degrees of freedom and using a significance level at of 5 for the probability ofa type I error toms 223 so we reject the null hypothesis Most regression software packages provide not only the OLS parameter estimates but the estimates of their standard deviations as well to facilitate tests of hypotheses about economic models The square of this tstatistic testing the signi cance of the explanatory variable in a bivariate regression equals the F1 10 statistic from Table 2 that tests whether the regression has explanatory power In a simple bivariate regression all of your hopes for explanation rest on a single independent variable so the tstatistic and the F statistic are linked


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