Nov 3 2005 LEC 12 ECON 240A 1 L Phillips Bivariate Normal Distribution Isodensity Curves I Introduction Economists 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 bivariate normal 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 isodensity lines or contour lines of the bivariate normal provide a visual representation The bivariate 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 3 2005 LEC 12 ECON 240A 2 L Phillips Bivariate Normal Distribution Isodensity Curves II Bivariate Normal Density The 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 2 x 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 one f 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 y x Figure 1 Isodensity Circles About the Origin Nov 3 2005 LEC 12 ECON 240A 3 L Phillips Bivariate Normal Distribution Isodensity Curves Note 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 2 exp 1 2 x2 1 2 exp 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 2 k 6 This is illustrated in Figure 2 y y x x Figure 2 Isodensity Lines About the Point of Means Bivariate Normal C Case 3 correlation is zero variance of x variance of y Nov 3 2005 LEC 12 ECON 240A 4 L Phillips Bivariate Normal Distribution Isodensity Curves If the variance of x exceeds the variance of y then the isodensity lines are ellipses about 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 2 For 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 y y Nov 3 2005 LEC 12 ECON 240A 5 L Phillips Bivariate Normal Distribution Isodensity Curves x x Figure 3 Isodensity Lines About the Point of the Means Var x Var y y y x x Figure 4 isodensity lines x and y correlated III Marginal Density Functions If 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 f x y dy 1 x 2 exp 1 2 x x x 2 10 Nov 3 2005 LEC 12 ECON 240A 6 L Phillips Bivariate Normal Distribution Isodensity Curves and similarly for y IV Conditional Density Function The 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 i e this is the expected value of y for a given value of x such as x E y x x y x x y x 12 So if x is at its mean x then the expected value of y is its mean y If x is above its mean and the correlation is positive then the expected value of y conditional on x is greater than y This is called the regression of y on x with …
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