# Ch2 - Chapter 2 Bivariate Distributions 2.1 Descriptive...

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0 50 100 150 200 250 300 350 400 Thousands 500 1000 1500 2000 2500 3000 3500 4000 Size (square feet) House Prices Vrs Size 1983-86 & 1987-88 Figure 2.1 Price Major Axis Chapter 2 Bivariate Distributions 2.1 Descriptive Statistics for Bivariate Distributions: Covariance and Correlation Chapter 1 focuses on the distribution of single random variables, such as the wage received by an individual worker. Empirical studies in economics typically seek to understand the relationship between two or more random variables. For example, how does the level of education affect the wage a worker can expect to earn? This chapter considers bivariate distributions; the statistical relationship between a pair of random variables. For example, if X and Y are both normally distributed random variables, their joint distribution is known as the bivariate normal distribution. Before exploring the properties of a theoretical distribution such as the bivariate normal, we will consider how descriptive statistics can be used to capture some essential features of the relationship between a pair of variables. Figure 2.1 is a scatter diagram of house prices and house sizes. Each point in the diagram corresponds to the price and size of a particular house. The sample consists of 2181 observations on house sales gathered over a 6 year period: 1983-88 (but the sample has no data for 1986.) Price refers to the 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 price - the scatter of points stretches up from the lower left towards the top right. It confirms what we would expect: larger houses tend to sell for higher prices. An interesting question that these data can answer is: by how much does the market price increase when size increases? Indeed the central goal of this chapter is to consider how we might frame and answer the question of the quantitative relationship between two variables such as the size and price of houses. Is the relationship linear or nonlinear? If it is linear, then what line “best” represents the

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Econometrics Text by D M Prescott © Chapter 2, 2 1 The scatter seems to be split into two concentrations, one above the other. The 1987-88 data presumably lie above the 1983-85 data, being separated by a gap created by the unrecorded 1986 data. Intercept ± ¯ Y ² slope × ¯ X ± \$(95510 ² 114.74×1313.5) ±² \$55201 size-price relationship? A natural choice for the line that captures the linear relationship between size and price is known as the major axis which is drawn in Figure 2.1. The key properties of the major axis are listed in Table 2.1. In particular, the slope of the major axis is the standard deviation of the Y-axis variable (price) divided by the standard deviation of the X-axis variable (size.) Table 2.1 Properties of the Major Axis 1. The slope is the ratio of standard deviations: SD(Y)/SD(X); Y is Price (P) and X is Size (S) 2. Passes through the sample mean point: ( ¯ X , ¯ Y ) 3. Bisects the scatter plot symmetrically The information in Table 2.2 lists univariate sample statistics for price and size and these can be
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Ch2 - Chapter 2 Bivariate Distributions 2.1 Descriptive...

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