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ECON 2740 (43)


30 Pages

Course Code
ECON 2740
David Prescott

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Chapter 2Bivariate Distributions21 Descriptive Statistics for Bivariate Distributions Covariance and CorrelationChapter 1 focuses on the distribution of single random variables such as the wage received by anindividualworkerEmpirical studies in economics typically seek to understand the relationship betweentwo or more random variablesFor example how does the level of education affect the wage a workercan expect to earnThis chapter considers bivariate distributions the statistical relationship between apair of random variablesFor example if X and Y are both normally distributed random variables theirjoint distribution is known as the bivariate normal distributionBefore exploring the properties of atheoretical distribution such as the bivariate normal we will consider how descriptive statistics can beused to capture some essential features of the relationship between a pair of variablesHouse Prices Vrs SizeFigure 21198386198788Price400 350 Major300 Axis250 200 Thousands150 100 50 0 500 1000 1500 2000 2500 3000 3500 4000 Size square feetFigure 21 is a scatter diagram of house prices and house sizesEach point in the diagramcorresponds to the price and size of a particular houseThe sample consists of 2181 observations onhouse sales gathered over a 6 year period 198388 but the sample has no data for 1986Price refers tothe price for which the house sold and size is the total floor space of the house measured in square feet The scatter plot reveals a positive relationship between size and pricethe scatter of points stretches upfrom the lower left towards the top rightIt confirms what we would expect larger houses tend to sellfor higher pricesAn interesting question that these data can answer is by how much does the marketprice increase when size increasesIndeed the central goal of this chapter is to consider how we mightframe and answer the question of the quantitative relationship between two variables such as the size andprice of housesIs the relationship linear or nonlinearIf it is linear then what line best represents theChapter 22Econometrics Text by D M Prescott sizeprice relationship A natural choice for the line that captures the linear relationship between size and price is knownas the major axis which is drawn in Figure 21 The key properties of the major axis are listed in Table21In particular the slope of the major axis is the standard deviation of the Yaxis variable pricedivided bythe standard deviation of the Xaxis variable size Table 21Properties of the Major Axis1The slope is the ratio of standard deviations SDYSDXY is Price P and X is Size SXY2Passes through the sample mean point 3Bisects the scatter plot symmetricallyThe information in Table 22 lists univariate sample statistics for price and size and these can beused to calculate the slope of the major axis which is 431133757 sq ft11474 per square foot To plot the major axis in Figure 21 it is necessary to find its linear equation slope and interceptTheintercept is chosen so that the major axis passes through the sample meanThe following formulaguarantees this resultInterceptYslopeX95510114741313555201The equation of the major axis in Figure 21 is therefore P552011147SAlthough Figure 21 shows a positive relationship between size and price the points are clearlydispersed around the major axisFor any particular size that is fixed on the Xaxis house prices span awide range because other variables affect the market price of housesThe age of a house its location anda host of other factors all play a part in determining market valueAn important variable is the date ofthe saleThese data were collected during the 1980s when property prices in Canada were rising and thatexplains some of the vertical spread in the plotfor any given size prices in 1988 will generally be much1higher than in 19831 The scatter seems to be split into two concentrations one above the otherThe 198788 datapresumably lie above the 198385 data being separated by a gap created by the unrecorded 1986 data
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