I. IntroductionII. Bivariate Normal DensityIII. Marginal Density FunctionsIV. Conditional Density FunctionV. Discriminating Between Two PopulationsNov. 9, 2006 LEC #12 ECON 240A-1 L. PhillipsBivariate Normal Distribution: Isodensity CurvesI. IntroductionEconomists rely heavily on regression to investigate the relationship between a dependent variable, y, and one or more independent variables, x, w, etc. As we have seen, graphical analysis often provides insight into these bivariate relationships and can reveal non-linear dependence, outliers, and other features that may complicate the analysis.There are other methodologies for examining bivariate relations. We have examined some of them. For example, correlation analysis, using the correlation coefficient, , is one method, as discussed in Lecture Eight. Another method is contingency table analysis. We will discuss the latter shortly. First we turn to the bivariatenormal distribution, which provides a useful visual model for bivariate relationships just as the univariate normal distribution provides a useful probability model for a single variable.It is useful to have a mental model in mind for bivariate relationships and the iso-density lines, or contour lines of the bivariate normal provide a visual representation. Thebivariate normal distribution of two variables, y and x, is a joint density function, f(x,y), and if the variables are jointly normal, then the marginal densities, e.g. f(x) and f(y), are each normal. In addition, the conditional densities, y given x, f(y/x), are normal as well.The isodensity lines, i.e. the locus where f(x,y) is constant, is a circle around the origin for the bivariate normal if both x and y have mean zero and variance one, i.e. are standardized normal variates, and are not correlated. If x and y have nonzero means, x and y , respectively, then these contour lines are circles around the point (x, y).If x has a larger variance than y, then the contour lines are ellipses with the long axis in the x direction. If x and y are correlated, then these ellipses are slanted.Nov. 9, 2006 LEC #12 ECON 240A-2 L. PhillipsBivariate Normal Distribution: Isodensity CurvesII. Bivariate Normal DensityThe density function, f(x,y) for two jointly normal variables, x and y where, for example, x has mean x, variance x2, and correlation coefficient , is:f(x, y) = 1/[2x y (1-2)] exp{(-1/[2(1-2)])([(x- x)/x]2 - 2[(x- x)/x ][(y- y)/y] +[(y- y)/y]2 }. (1)A. Case 1: correlation is zero, means are zero, and variances are onef(x, y) = 1/[2 ] exp{(-1/2)[ x2 + y2 ]} (2)and for an isodensity, where f(x,y) is a constant, k, taking logarithms,ln [2 f(x, y)] = -1/2 [x2 + y2 ],or [x2 + y2 ] = -2 ln [2 f(x, y)] = -2ln [2 k]. (3)Recall [x2 + y2] = r2 is the equation of a circle around the origin, (0, 0) with radius r, as illustrated in Figure 1.-------------------------------------------------------------------------------- Figure 1: Isodensity Circles About the OriginyxNov. 9, 2006 LEC #12 ECON 240A-3 L. PhillipsBivariate Normal Distribution: Isodensity CurvesNote that if x and y are independent, then the correlation coefficient, , is zero and the joint density function, f(x, y), is the product of the marginal density functions for x and y, i.e. f(x, y) = f(x) f(y) = 1/2exp [-1/2 x2 ] 1/2exp [-1/2 y2 ] (4)where x and y have mean zero and variance one.B. Case 2: correlation is zero, variances are one, means x and y In this case, the origin is translated to the point of the means, (x, y). The bivariate density function is:f(x, y) = 1/(2) exp {(-1/2)[(x - x)2 + (y - y)2 ]}. (5)For a density equal to k:[(x - x)2 + (y - y)2 ] = -2 ln [2 f(x,y)] = -2 ln[2k] (6)This is illustrated in Figure 2.-------------------------------------------------------------------------------------Figure 2: Isodensity Lines About the Point of Means, Bivariate Normal C. Case 3: correlation is zero, variance of x > variance of yyxxyNov. 9, 2006 LEC #12 ECON 240A-4 L. PhillipsBivariate Normal Distribution: Isodensity CurvesIf the variance of x exceeds the variance of y, then the isodensity lines are ellipsesabout the point of the means with the semi-major axis in the x direction:f(x,y) = 1/(2 x y ) exp{ (-1/2) ([(x-x)/x]2 + [(x-y)/y]2 )} (7)Note that if x and y are independent, then the correlation coefficient is zero and the joint density is the product of the marginal densities:f(x, y) = f(x) f(y) = 1/(x 2) exp[-1/2[(x- x)/x]2 1/(y 2) exp[-1/2[(y- y)/y]2For a constant isodensity, f(x, y) = k, from Eq. (7) we have,([(x-x)/x]2 + [(x-y)/y]2 = -2 ln (2 x y f(x, y)) = -2 ln (2 x y k) (8)Recall the equation of an ellipse about the origin with semi-major axis a and semi-minor axis b is:x2/a2 + y2/b2 = 1 (9)Elliptical isodensity lines around the point of the means are illustrated for Eq. (7) in Figure 3.Case 4: correlation is nonzero.The joint density function is given by Eq. (1) above, and the isodensity lines are tilted ellipses around the point of the means as illustrated in Figure 4, for positive autocorrelation.yyNov. 9, 2006 LEC #12 ECON 240A-5 L. PhillipsBivariate Normal Distribution: Isodensity Curves Figure 3: Isodensity Lines About the Point of the Means, Var x > Var y-----------------------------------------------------------------------------------Figure 4: isodensity lines, x and y correlated----------------------------------------------------------------------------------------------III. Marginal Density FunctionsIf x and y are jointly normal, then both x and y each have normal density functions. For example, the marginal density of x, f(x) is:f(x) = dyyxf ),(= 1/(x 2) exp[-1/2[(x- x)/x]2 (10) xxxxyyNov. 9, 2006 LEC #12 ECON 240A-6 L. PhillipsBivariate Normal Distribution: Isodensity Curvesand similarly for y.IV. Conditional Density FunctionThe density of y conditional on a particular value of x, x = x*, is just a vertical slice of the isodensity curve plot at that value of x, and if x and y are jointly normal, is also normal. It can be obtained by dividing the joint density function by the marginal density and simplifying:f(y/x) = f(x, y)/f(x) = 1/[y 2(1 - 2)1/2] exp{[-1/[2(1-2)y2][y-y-(x-x)(y/x)]}(11)where the mean of the conditional distribution is y + (x-x)(y/x),
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