ࡱ> M pbjbj== WWblt%t%t%8%@&hS0'f."../2vY3,3RRRRRRR$T >VRQ3I2233RqF./E"SqFqFqF3 .L/RqF3RqFqFwN@wN/$' R`M#t%@wNwN48S0hSwNVqFVwNqFDerivation of the Ordinary Least Squares Estimator Simple Linear Regression Case As briefly discussed in the previous reading assignment, the most commonly used estimation procedure is the minimization of the sum of squared deviations. This procedure is known as the ordinary least squares (OLS) estimator. In this chapter, this estimator is derived for the simple linear case. The simple linear case means only one x variable is associated with each y value. Simple Linear Regression Error Defined The simple linear regression problem is given by the following equation (1)  EMBED Equation.3  where yi and xi represent paired observations, a and b are unknown parameters, ui is the error term associated with observation i, and n is the total number of observations. The terms deviation, residual, and error term are often used interchangeably in econometric analysis. More correct usage, however, is to use error term to represent an unknown value, residual, deviation, or estimated error term to represent a calculated value for the error term. The term a represents the intercept of the line and b represents the slope of the line. One key assumption is the equation is linear. Here, the equation is linear in y and x, but the equation is also linear in a and b. Linear simply means nonlinear terms such as squared or logarithmic values for x, y, a, and b are not included in the equation. The linear in x and y assumption will be relaxed later, but the equation must remain linear in a and b. Experience suggests this linear requirement is an obstacle for students understanding of ordinary least squares (see linear equation review box). You have three paired data (x, y) points (3, 40), (1, 5), and (2, 10). Using this data, you wish to obtain an equation of the following form (2)  EMBED Equation.3  where i denotes the observation, a and b are unknown parameters to be estimated, xi is the independent variable, yi is the dependent variable, and ui is the error term associated with observation i. Simple linear regression uses the ordinary least squares procedure. As briefly discussed in the previous chapter, the objective is to minimize the sum of the squared residual,  EMBED Equation.3 . The idea of residuals is developed in the previous chapter; however, a brief review of this concept is presented here. Residuals are how far the estimated y (using the estimated y-intercept and slope parameters) for a given x is from an observed y associated with the given x. By this definition, residuals are the estimated error between the observed y value and the estimated y value. Included in this error term is everything affecting y, but not included in the equation. In the simple linear regression case, only x is included in the equation, the error term includes all other variables affecting y. Graphically, residuals are the vertical distance (x is given or held constant) from the observed y and the estimated equation. Residuals are shown graphically in figure 1 (as before figures are at the end of this reading assignment). In this figure, residuals one and two are positive, whereas, residual three is negative. For residual one, the observed y value is 40, but the estimated y value given by the equation is 35.83. The residual associated with this data point is 4.17. Recall, residuals are calculated by subtracting the estimated value from the observed value (40 - 35.83). In general terms, residuals are given by the equation  EMBED Equation.3  where the hat symbol denotes an estimated y value (the value given by the equation). Note the subscript on  EMBED Equation.3 , as before this subscript denotes observation i. An estimated error is calculated for each observation. This allows the information in each observation to be used in the estimation procedure. Recall, from earlier readings, this is an important property of OLS. To recap, using the above example, we have the following information, 1) paired observations on y and x, 2) the total number of observations is n, 3) a simple functional form given by yi = a + bxi + ui, 4) a and b are unknown parameters, but are fixed (that is they do not vary by observation), 5) a definition for an error term, 6) the objective to minimize the sum of squared residuals, 7) one goal to use all observations in the estimation procedure, and 8) we will never know the true values for a and b. Minimize Sum of Squared Errors To minimize the sum of squared residuals, it is necessary to obtain  EMBED Equation.3  for each observation. Estimated residuals are calculated using the estimated values for a and b. First, using the estimated equation EMBED Equation.3 , (recall the hat denotes estimated values) an estimated value for y is obtained for each observation. Unfortunately, at this point we do not have values for  EMBED Equation.3 . As noted earlier, residuals are calculated as  EMBED Equation.3 . At this point, we have values for all yis. The sum of squared residuals can be written mathematically as (3)  EMBED Equation.3  where n is the total number of observations and " is the summation operator. The above equation is known as the sum of squared residuals (sum of squared errors) and denoted SSE. Using the definitions of  EMBED Equation.3  and  EMBED Equation.3 , the SSE becomes (4)  EMBED Equation.3  where first the definition for  EMBED Equation.3  is used, then the definition for  EMBED Equation.3 , and finally some algebra. Up to this point, only algebra has been used to provide a definition for residuals and provide an equation for the sum of squared residuals. Because the objective is to minimize the sum of squared residuals, a procedure is necessary to achieve this objective. Recall from the calculus review, a well-behaved function can be maximized or minimized by taking the first derivatives and setting them equal to zero. Second order conditions are then checked to determine if a maximum or minimum has been found. An equation of the objective function for the OLS estimation procedure is: (5)  EMBED Equation.3  with respect to (w.r.t.)  EMBED Equation.3 . KEY POINT: Students at this point often face a mental block. Minimizing this equation w.r.t.  EMBED Equation.3  is no different than minimizing equations seen in the review of calculus. The mental block occurs because earlier, the minimization was w.r.t. x and not the parameters,  EMBED Equation.3 and  EMBED Equation.3 . Calculus does not depend on what the variable of interest is called. Do not change the procedure and ideas, just because the variables of interest are now called  EMBED Equation.3 and  EMBED Equation.3 and not x. This is simply a nomenclature issue. Estimates for a and b are obtained by minimizing the SSE as follows 1) take the first order partial derivatives w.r.t.  EMBED Equation.3 , 2) set the resulting equations equal to zero, 3) solve the first order conditions (FOC) for  EMBED Equation.3 , and 4) check the second order conditions for a maximum or minimum. Notice this procedure is no different than the procedure reviewed earlier. Taking the first order partial derivatives w.r.t.  EMBED Equation.3 and setting them equal to zero, the following two equations are obtained: (6)  EMBED Equation.3  Two equations with two unknowns,  EMBED Equation.3 , are obtained from the first order conditions. Solving these equations for  EMBED Equation.3 one obtains the following expressions for the values of  EMBED Equation.3 that minimize the SSE. From the first order conditions (equation 6), the following general expressions are obtained for the OLS estimators (see tables 1 and 2 for steps involved): (7)  EMBED Equation.3 . Table 1. Algebraic Steps Involved in Obtaining the OLS Estimator,  EMBED Equation.3 , from the FOC ConditionsMathematical DeviationStep involves EMBED Equation.3 Original FOC EMBED Equation.3 Divide both sides by -2 EMBED Equation.3 Distribute the summation operator EMBED Equation.3 Summation over a constant EMBED Equation.3 Subtraction EMBED Equation.3 Divide by n EMBED Equation.3 Definition of a mean Table 2. Algebraic Steps Involved in Obtaining the OLS Estimator,  EMBED Equation.3 , from the FOC ConditionsMathematical DeviationStep involves EMBED Equation.3 Original FOC EMBED Equation.3 Divide both sides by -2 EMBED Equation.3 Distribute the summation operator and xi EMBED Equation.3 Summation over a constant EMBED Equation.3 Substitute the definition for the  EMBED Equation.3  EMBED Equation.3 Use the distributive law and then factor out  EMBED Equation.3 , careful with the negative and positive signs EMBED Equation.3 Solve for  EMBED Equation.3  EMBED Equation.3 Simplify Example. At this point, returning to the example will help clarify the procedure. There are three paired observations on x and y, (3, 40), (1, 5), and (2, 10). Using these data points, the SSE can be written as (8)  EMBED Equation.3  Taking the partial derivative of equation (7) w.r.t.  EMBED Equation.3 and setting the resulting equations equal to zero, the following first order conditions are obtained: (9)  EMBED Equation.3  Solving these two equations for the two unknowns,  EMBED Equation.3 , one obtains the OLS estimates for a and b given the three paired observations. The first equation in equation 9 can be simplified as: (10)  EMBED Equation.3  Similarly, the second equation in equation (6) can be simplified as: (11)  EMBED Equation.3  One way of solving these two equations is to multiplying equation (10) by 2 and then setting equations (10) and (11) equal to each other. Another way is to solve one equation for  EMBED Equation.3  or  EMBED Equation.3  and substitute into the other equation. Both methods will give the same answer. Solving the equations, one obtains the OLS estimate for a (note both equations are equal to zero and multiplying both sides by two will not affect the equality):  EMBED Equation.3  Substituting this result into either equation (10) or (11), the OLS estimate for a is obtained:  EMBED Equation.3  The OLS estimates for this example are, therefore,  EMBED Equation.3 . Given the original simple linear equation and the three data points, an intercept of -16.67 and a slope of 17.5 are obtained; giving the following estimated line  EMBED Equation.3 . The following information can be in the general formulas to obtain the OLS estimates. These general formulas, avoid having to solve the FOC using the data observations. yixixiyix2i40312095151102204Summation55614514Mean18.3332Using this information, the following equations are obtained: (12)  EMBED Equation.3  Using the estimated equation, estimated errors for each observation are: (12)  EMBED Equation.3  Second Order Conditions As with all maximization and minimization problems, the second order conditions (SOC) need to be checked to determine if the point given by the FOC conditions is a maximum or minimum. The OLS problem will provide a minimum. Intuitively, a minimum is achieved because the problem is to minimize the sum of squares. Squaring each residual forces each term to be nonnegative. The problem becomes then minimizing the sum of nonnegative numbers. Because all numbers are nonnegative, the smallest possible sum is zero. A zero sum occurs when all residuals equal zero. This would be a perfect between the line and the data points. In empirical studies, a perfect will not occur. A residual that is positive will add to the sum of the squares. Thus, the sum of squared residuals must equal a zero or a positive number. This intuitive explanation indicates the problem will provide a minimum. In figure 2, the general minimization problem is shown graphically (note, in the figure the intercept is b1 instead of a and the slope parameter is b2, do not let the change in notation confuse you. An unknown variable can be called anything. This is the only good figure I could find). For different estimates for a and b, the SSE is graphed. Notice the SSE varies as the estimates for the intercept and slope change. Of importance here is the shape of the minimization problem. Notice, the bowl shaped function. Such a shape assures the SOC will be satisfied. Mathematically, the FOC for minimization of the SSE finds the lowest point on the SSE graph. The SOC confirm the point is a minimization. To use the simple second order condition test from the calculus review section of the class, the second and cross partial derivatives are necessary. Recall, the incomplete test required for a minimum the second order partials to be greater than zero and the two cross partials multiplied together must be greater than the cross partials squared. The necessary partial derivatives are: (13)  EMBED Equation.3  The second order partials are greater than zero, because only nonnegative numbers are involved. An observation on the variable, x, maybe negative, but squaring x results in a positive number. Further, it can be shown (although not intuitively) that  EMBED Equation.3 . This equation holds partially because one side of the equation is multiplied by the number of observations. Example Continued. Continuing the example, it is necessary to check the second order conditions. Recall, the three-paired observations on x and y are (3, 40), (1, 5), and (2, 10). Using these data points, the SOC are:  EMBED Equation.3  Therefore, the second order conditions hold and the OLS estimates minimized the SSE w.r.t.  EMBED Equation.3 . In figure 3, the minimization problem is shown graphically for our simple three-observation example. For different estimates for a and b, the SSE is graphed. Notice the SSE varies as the estimates for a and b change. As the estimates for a and b move away from the OLS estimates of -16.67 and 17.5, the SSE increases. KEY POINT: although often seen as using new ideas, the derivation of the OLS estimator uses only simple algebra and the idea of minimization of a quadratic function. This is material that was covered in the prerequisites for this class and reviewed in previous lectures. What is new is the combining of the subject matters of economics (for the problem), algebra (model set up), and calculus (minimization) to obtain the OLS estimates for a simple linear equation. Another point other seen as a problematic is that in previous classes, algebra was used to find x and y with a and b as given constants. In econometrics, the xs and ys are given constants and estimates for a and b are obtained. KEY POINT: two assumptions are implicit in the previous derivation of the OLS estimator. First, the original problem assumed the equation to be estimated was linear in a and b. Second, it was assumed the FOC could be solved. Neither assumption is particularly restrictive. Notice, the assumptions say nothing about the statistical distribution of the estimates, just that we can get the estimates. One reason OLS is so powerful is that estimates can be obtained under these fairly unrestrictive assumptions. Because the OLS estimates can be obtained easily, this also results in OLS being misused. The discussion will return to these assumptions and additional assumptions as we continue deriving the OLS estimator. Algebraic Properties of the OLS Estimator Several algebraic properties of the OLS estimator are shown here. The importance of these properties is they are used in deriving goodness-of-fit measures and statistical properties of the OLS estimator. Algebraic Property 1. Using the FOC w.r.t.  EMBED Equation.3 it can be shown the sum of the residuals is equal to zero: (14)  EMBED Equation.3  Here, the first equation is the FOC derived earlier. The definition for the residuals  EMBED Equation.3  is substituted into the FOC equation. The constant, -2, is then removed from the summation. Because the constant 2 does not equal zero, the only way the FOC can equal zero is the sum of the residuals equal zero. The sum of the residuals equally zero, implies the mean of the residuals must also equal zero. Algebraic Property 2. The point  EMBED Equation.3 will always be on the estimated line. Using the FOC w.r.t  EMBED Equation.3 , this property can be shown by dividing both sides by -2, distributing the summation operator through the equation, divided both sides by the number of observations, and then simplifying by using the definition of a mean. Mathematically this property is derived as follows, (15)  EMBED Equation.3  Algebraic Property 3. The sample covariance between the xi and  EMBED Equation.3  is equal to zero. Recall, the formula for calculating the covariance between any two variables, z and q is  EMBED Equation.3 . To show this property, the FOC w.r.t. EMBED Equation.3 is used. By substituting in the definition for the residuals, the FOC can be simplified to: (16)  EMBED Equation.3  Substituting this result, along with the algebraic property 1, the sum and mean of the residuals equal zero into the covariance formula, algebraic property 3 is derived: (17)  EMBED Equation.3  This shows the covariance between the estimated residual and the independent variable is equal to zero. Algebraic Property 4. The mean of the variable, y, will equal the mean of the  EMBED Equation.3 . Distributing the summation operator through the definition of the estimated residual, using algebraic property 1, and the definition of a mean are used to show this property. Mathematically, this is property is derived as: (18)  EMBED Equation.3 . Example Continued. Using the previous three-observation example, it can be shown the four algebraic properties hold. Algebraic Property 1. Using the residuals calculated earlier, the sum of the residuals is SSE = 4.17 + 4.17 - 8.33 = .01 which within rounding error equals zero. Algebraic Property 2. Substituting in the mean of the xs, 2, into the estimated equation, one obtains y = -16.65 + 17.5 (2) = 18.33, which equals the mean of the y variables, 18.33 = (40 + 5 + 10)/3. Algebraic Property 3. The covariance between x and  EMBED Equation.3 is given by  EMBED Equation.3  Thus, property 3 holds. Algebraic Property 4. Using earlier calculations for obtaining the estimated ys, the mean of the estimated ys can be obtained using the equation (35.83 + 0.83 + 18.33)/3 = 18.33. Thus, the mean of y and  EMBED Equation.3 are equal. KEY POINT: in deriving these algebraic points, subject matter (definitions of mean and covariance) that pertains to statistics has been added to the mix of economics, calculus, and algebra. Goodness-of-Fit Up to this point, nothing has been stated about how good the estimated equation fits the observed data. In this section, one measure of the goodness-of-fit is presented. This measure is called the coefficient of determination or R2. The coefficient of determination measures the amount of the sample variation in y that is explained by x. To derive the coefficient of determination, three definitions are necessary. First, the total sum of squares (SST) is defined as the total variation in y around its mean. This definition is very similar to that of a variance. SST is defined as (19)  EMBED Equation.3 . Notice, this formula is the same, as the formula for a variance except the variation is not divided by the degrees of freedom. We will return to this point later in the lectures. The second need is the explained sum of squares (SSR) or sum of squares of the regression (where the R comes from). SSR is simply the variation of the estimated ys around their mean. SSR is defined as: (20)  EMBED Equation.3 . Recall, the actual values for y and the estimated values for y have the same mean. Therefore, equations (19) and (20) are the same except for actual or estimated ys are used. The means are the same. The third definition is the residual sum of residuals (errors) (SSE), which was earlier defined as: (21)  EMBED Equation.3  EMBED Equation.3  SSE is the amount of variation not explained by the regression equation. It be shown the total sum of the variation in y around its mean is equal to the amount of variation in y around its mean plus the amount of variation not explained. Mathematically, this statement is SST = SSR + SSE. To show this equation holds, algebraic properties (1) and (3) derived earlier must be used. Using these two properties, and expanding the SST equation, the necessary steps to show this equation holds are shown in Table 3. Table 3. Algebraic Steps Involved in Showing the Equation SST = SSR + SSE HoldsMathematical DeviationStep involves EMBED Equation.3 Original equation EMBED Equation.3 Add  EMBED Equation.3 and subtract  EMBED Equation.3  EMBED Equation.3 Use the definition  EMBED Equation.3  EMBED Equation.3  Expand EMBED Equation.3 Distribute the summation operator EMBED Equation.3 Using sum of square definitions  EMBED Equation.3 Need to show middle term equals zero EMBED Equation.3 Use definition  EMBED Equation.3  EMBED Equation.3 Divide by 2 and distribute the summation operator EMBED Equation.3 Using algebraic properties 1 and 3SST = SSR + SSEEquation shown From the equation SST = SSR + SSE, the coefficient of determination can be derived. Taking this equation and dividing both sides by SST one obtains: (22)  EMBED Equation.3  This equation is equal to one because any number divided by itself equals one. Rearranging this equation the coefficient of determination is obtained: (23)  EMBED Equation.3  As shown in this equation, the coefficient of determination, R2, is the ratio of the amount of variation explained to the total variation in y around its mean, or equivalently, the one minus the ratio of the amount of variation not explained to the total variation. Thus, R2 measures the amount of sample variation in y that is explained by x. R2 can range from zero (no fit) to one (perfect fit). If x explains no variation in y, the SSR will equal zero. Looking at equation (23), a zero for SSR gives a value of zero for R2. On the other hand, if x explains all the variation in y, SSR will equal SST. In this case, R2 equals one. The values of [0 - 1] are just the theoretical range for the coefficient of determination. One will not usually see either of these values when running a regression. The coefficient of determination (and its adjusted value discussed later) is the most common measure of the fit of an estimated equation to the observed data. Although, the coefficient of determination is the most common measure, it is not the only measure of the fit of an equation. One needs to look at other measures of fit, that is dont use R2 as your only gauge of the fit of an estimated equation. Unfortunately, there is not a cutoff value for R2 that gives a good measure of fit. Further, in economic data it is not uncommon to have low R2 values. This is a fact of using socio-economic cross-sectional data. We will continue the discussion on R2 later in this class, when model specification is discussed. Example Concluded. To finish our example, we need to calculate SST, SSR, and SSE, along with the coefficient of determination. Using the residuals calculated earlier, SSE equals SSE = (4.17)2 + (4.17)2 + (-8.33)2 = 104.17. The estimated y values calculated and means earlier are used to obtain SSR: SSR = (35.83 - 18.33)2 + (0.83 - 18.33)2 + (18.33 - 18.33)2 = 612.5. Using the actual observations on y, SST equals: SST = (40 - 18.33)2 + (5 - 18.33)2 + (10 - 18.33)2 = 716.67. Thus, the coefficient of determination is R2 = 612.5 / 716.67 = 0.85 or R2 = 1 - (104.17 / 716.67) = 0.85. In our example, the amount of variation in y around its mean explained by the estimated equation is 85%. Important Terms / Concepts Error Term Residual Deviation Estimated error term Hat symbol Sum of squares SSR SST SSE Why OLS is powerful? Why OLS is misused? Four Algebraic properties Goodness-of-fit R2 - Coefficient of Determination Range of R2 n i   EMBED Excel.Chart.8 \s  Figure 2. General simple linear regression problem of minimizing the sum of squared errors w.r.t. the intercept (b1) and slope (b2) parameters. Note in the notation in the text, a denotes the intercept and b denotes the slope.  Figure 3. Graph showing the minimization of the sum of squared errors for the simple linear regression example  PAGE  PAGE 1 Linear Equation Review Linear equations are simply equations for a line. Recall, from algebra, an equation given by y =  + x has a y-intercept equal to  and a slope equal to . The y-intercept gives the point where the line crosses the y-axis. If the intercept is 10, the line will cross the y-axis at this value. Crossing the y-axis indicates a x-value of zero, therefore, the (x, y) point (0, 10) is associated with this line. The slope indicates the rise and run of the line. A slope equal to five indicates y increases by five units (the line rises five units) for each one-unit increase in x (a one unit increase in x). A positive slope indicates the line is upward sloping, whereas a negative slope indicates a downward sloping line. Which of the following equations are linear in x and y?  EMBED Equation.3  Only the first equation is linear in y and x. The second equation contains an x-squared term. This equation is a quadratic equation. The third equation is also is not a linear equation in x and y; here the natural logarithm of x is in the equation. This equation is commonly known as a semi-log equation. However, each equation remains linear in the y-intercept and slope parameters. Of the following equations, which one(s) are linear in the y-intercept and slope parameters?  EMBED Equation.3  Here, none of the equations is linear in the y-intercept and slope parameters. Each equation contains a nonlinear component associates with at least one of the parameters. 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FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjffObjInfohEquation Native i|_1136741003F _M _M.`mIyI 2a ="16.67and2b =17.5 FMicrosoft Equation 3.0 DS Equation Equation.39qH0.)" 2 SSR"2b  2 =2x i2i=1n " =2x i2i=1n " >0)" 2 SSR"2a "2b =2x ii=1n " =2x ii=1n " . FMicrosoft Equation 3.0 DS Equation Equation.39qۀmIyI 2n(2x i2 " )e"(2x iOle CompObjfObjInfoEquation Native  " ) 2 FMicrosoft Equation 3.0 DS Equation Equation.39q(I(I )" 2 SSR"2a  2 =2 i=1n " =2n=2(3)=6>0_1134880793F_M_MOle CompObjfObjInfo Equation Native D_1134881210&#F_M_MOle CompObj"$f.)" 2 SSR"2b  2 =2x i2i=1n " =2x i2i=1n " =2(3 2 +1 2 +2 2 )=28>0)" 2 SSR"2a  2 ()" 2 SSR"2b  2 )>()" 2 SSR"2a "2b ) 2 !6(28)>(2(3+1+2)) 2 !168>144I FMicrosoft Equation 3.0 DS Equation Equation.39q0mIyI 2a and2ObjInfo%Equation Native L_1134884422(F`_M`_MOle b  FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI 2a  FMicrosoft Equation 3.0 DS EqCompObj')fObjInfo*Equation Native 0_1134884752!5-F`_M`_MOle CompObj,.fObjInfo/Equation Native uation Equation.39qmIyI "2(y i "2a "2b x i )=0 i " "22u  i =0!2u  i =0. i=1n " i "_11348847932F`_M`_MOle CompObj13fObjInfo4 FMicrosoft Equation 3.0 DS Equation Equation.39qhIoI (2u  i =y i "2a "2b x i ) FMicrosoft Equation 3.0 DS EqEquation Native _11348850370:7F`_M`_MOle CompObj68fuation Equation.39q,mIyI (2y,2x) FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo9Equation Native H_1134885428<F`_M`MOle CompObj;=fObjInfo>Equation Native l_1134885719+gAF`M`MPI(I "2(y i "2a "2b x i )=0 i " y i " i " 2a " i " 2b x i =y i "n2a "2b x ii " = i " 0 i " y i " n"2a "2b x i " n=0n2y"2a "2b 2x=02y=2a +2b 2x. FMicrosoft Equation 3.0 DS Equation Equation.39q mIyI 2u  iIOle CompObj@BfObjInfoCEquation Native <_1134886059FF`M`MOle CompObjEGfObjInfoH FMicrosoft Equation 3.0 DS Equation Equation.39qۜI(I cov=(z i "2z)(q i "2q) i=1n " n"1 FMicrosoft Equation 3.0 DS EqEquation Native _1134886720DNKF`M`MOle CompObjJLfuation Equation.39qmIyI 2b  FMicrosoft Equation 3.0 DS Equation Equation.39qI(I "2(y ObjInfoMEquation Native 0_1134886811PF`M@`MOle CompObjOQfObjInfoREquation Native _1134887270I]UF@`M@`Mi "2a "2b x i )x i =0 i " 2u  i x i =0 i " FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI cov(x,2Ole CompObjTVfObjInfoWEquation Native    "%&'(),1458;<?BEJMPSTW\adehknqrsvyz{~u )=(x i "2x)(2u  i "2u )  " n"1=x i u i "x i 2u   "  " "2x2u  i +2x2u   "  " n"1=x i u i "2u x i "  " "2x2u  i +n2x2u   " n"1=0"0x i "2x0+n2x0  " n"1=0. FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI 2y _1134887629ZF@`M@`MOle  CompObjY[ fObjInfo\Equation Native 0_1134888016Xb_F@`M@`MOle CompObj^`f FMicrosoft Equation 3.0 DS Equation Equation.39qېmIyI 2u  i =y i "2y  i y i =2y  i +u i y i " =2y  i " +ObjInfoaEquation Native _1134888841dF`{`M`{`MOle 2u  i " y i " n=2y  i " n2y=2y . FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI 2u CompObjcefObjInfofEquation Native 0_1134889236SiF`{`M`{`MOle  CompObjhj!fObjInfok#Equation Native $h FMicrosoft Equation 3.0 DS Equation Equation.39qLII (3"2)(4.17"0)+(1"2)(4.17"0)+(2"2)("8.33"0)3"1=1(4.17)"1(4.17)"0("8.33)2=0. FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI 2y _1134889416nF`{`M`{`MOle *CompObjmo+fObjInfop-Equation Native .0_1134905025lsF`{`M `MOle /CompObjrt0f FMicrosoft Equation 3.0 DS Equation Equation.39qlmIyI SST=(y i "2y) 2i=1n " FMicrosoft Equation 3.0 DS EqObjInfou2Equation Native 3_1136742037xF `M `MOle 6CompObjwy7fObjInfoz9Equation Native :_1136742065}F `M `Muation Equation.39qpI(I SSR=(2y  i "2y) 2i=1n " FMicrosoft Equation 3.0 DS Equation Equation.39qOle =CompObj|~>fObjInfo@Equation Native AtXIhI SSE=2u  i2i=1n " . FMicrosoft Equation 3.0 DS Equation Equation.39qmIyI _1134905505F `M `MOle CCompObjDfObjInfoF  Equation Native G$_1135074591qF `M@D(`MOle HCompObjIf FMicrosoft Equation 3.0 DS Equation Equation.39qI\I(I SST=(y i "2y) 2 " FMicrosoft Equation 3.0 DS EqObjInfoKEquation Native Lx_1136742143{F@D(`M@D(`MOle NCompObjOfObjInfoQEquation Native R_1136742160F@D(`Ml1`Muation Equation.39qݔI(I =[(y i "2y  i )+(2y  i "2y)] 2 " FMicrosoft Equation 3.0 DS Equation Equation.39qOle UCompObjVfObjInfoXEquation Native Y< IhI 2y  i FMicrosoft Equation 3.0 DS Equation Equation.39q I`I 2y  i_1136742170Fl1`Ml1`MOle ZCompObj[fObjInfo]Equation Native ^<_1136742180Fl1`Ml1`MOle _CompObj`f FMicrosoft Equation 3.0 DS Equation Equation.39qtII =[2u  i +(2y  i "2y)] 2 " FMicrosoft Equation 3.0 DS EqObjInfobEquation Native c_1135074687Fl1`Ml1`MOle fCompObjgfObjInfoiEquation Native jl_1135074788Fl1`M 9`Muation Equation.39qIPmIyI 2u  i =y i "2y  i FMicrosoft Equation 3.0 DS Equation Equation.39qOle lCompObjmfObjInfooEquation Native pI٨I(I =[2u  i2 +22u  i (2y "2y)+(2y "2y) 2 ]  " FMicrosoft Equation 3.0 DS Equation Equation.39q_1136742220F 9`M 9`MOle tCompObjufObjInfowEquation Native x_1136742246F 9`M 9`MOle |CompObj}fII =2u  i2 +22u  i " (2y  i "2y)+(2y  i "2y)  "  2 " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1135075339F 9`M 9`MOle ݄LII =SSE+22u  i (2y  i "2y)  " +SSR FMicrosoft Equation 3.0 DS Equation Equation.39qIxI(I 22u  iCompObjfObjInfoEquation Native _1135075493F 9`M 9`M (2y  i "2y)  " '=  ?0t FMicrosoft Equation 3.0 DS Equation Equation.39qI}IXtI 2[2u  i (2a +2b x i "2y)Ole CompObjfObjInfoEquation Native 8=2[2u  i 2a +2u  i 2b x i "2u  i 2y]  "  " '=  ?0 FMicrosoft Equation 3.0 DS Equation Equation.39qIHdII 2y  i =2a_1135075442F4B`M4B`MOle CompObjfObjInfoEquation Native d_1136742283F4B`M4B`MOle CompObjf +2b x i FMicrosoft Equation 3.0 DS Equation Equation.39q(J0I 2a 2u  i " +2b 2u  i x i " "2y2u  i "ObjInfoEquation Native _1136742299FI`MI`MOle  '=  ?0 FMicrosoft Equation 3.0 DS Equation Equation.39q@8IJ 2a 0+2b 0"2y0=0CompObjfObjInfoEquation Native \_1136742334FI`MI`MOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q݀I,$J )SSTSST=)SSRSST+)SSESST=1. FMicrosoft Equation 3.0 DS Eq_1136742346FI`MI`MOle CompObjfObjInfouation Equation.39qtIJ R 2 =)SSRSST=1")SSESST. !FMicrosoft Excel ChartBiff8Excel.Chart.89qEquation Native _1136742828!FI`MI`MOle PRINT&9=2  X   ''  Times New Roman-Arial----Arial-"System-'- -'- ^-'-  "-+z z-nz,n,znzlnlz n znzKnKznznz+n+zzzzHHLL "--'-- --'-- (!p--'-- %!p_2- "- <%L26Oha!8ckE( w,c<QKAX3d&nv|<%|   {v%o/g;]GSUHd<t/!-Rydku]OC7,L- VB---'--- %!p| r ---'--- %!p, 6"---'--- %!pL- "-   $LVLBL---'---  %!p, $",6 ,"---'---  %!p $---'---  %!p---'---  (!p---'---  -------'---    X2 6Figure 1. 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Chart2Sheet1*Sheet2+Sheet3`iZR3  @@  Xy valuesx valuesSUMMARY OUTPUTRegression Statistics Multiple RR SquareAdjusted R SquareStandard Error ObservationsANOVA RegressionResidualTotal InterceptdfSSMSFSignificance F Coefficientst StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% X Variable 1y hat" 0})  A@MRHP OfficeJet G Series|݀ fdںںRLdͫLPT1:"d??3` Y ` Y ` Y ` % пl3d 3Q  EstimatedQ ;Q ;Q3_ ! NM ] !!d4E4 3Q ObservedQ ;Q ;Q3_4E4D$% MP+3O& Q4$% MP+3O& Q4FAK  3OJ=| 3*#M& 43*#M& 43d" r3O% Mp73OQ443_ M NM  MM<444% x* X M3O 2& Q OFigure 1. Relationship Between Estimated Equation and Observed Value, Residuals'44e@@??@@eA@D@?@UUUUUU2@$@e>  A@  dMbP?_*+%MRHP OfficeJet G Series|݀f dںںRLdͫLPT1:"d??U!                              D@@'A@DD D@?'?DD D$@@'UUUUUU2@DD D    OYI? ֔5eMY? )kʚ? i$@ ~ @          ?$@$@`@F_B?   ~ ? 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