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lecturenotes09 - ECON 41 Statistics for Economists Lecture...

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ECON 41: Statistics for Economists Lecture Notes 0 9 Chunming Yuan [email protected] Economics Department, UCLA
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In this lecture we will discuss tools for analyzing the relationship between two random variables: The correlation coe cient tells us how strongly and in which direction two variables are related. The (linear) regression model tells us how much a variable is expected to change due to a certain change in another variable. The values of two random variables can be graphically represented in a so-called scatterplot or scatter diagram which is a plot of paired observations.
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The Linear Correlation Coe cient The (linear) correlation coe cient measures the strength of the linear associ- ation between two random variables. We distinguish between the population correlation coe cient ,denotedbythe Greek letter ρ, and the sample correlation coe cient ,denotedby r. Both the population and sample correlation coe cients take values only be- tween -1 and 1, i.e. 1 ρ 1 1 r 1 Ava lueo f r (or of ρ ) equal to 1 means that there is a perfect positive linear relationship . If we plot the values of the two variables on the x and y axis in a so-called scatterplot
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Ava lueo f r (or of ρ ) equal to -1 means that there is a perfect positive linear relationship . If we plot the values of the two variables on the x and y axis in a so-called scatterplot (or scatter diagram) then they lie on a straight line with a negative slope. Ava lueof r (or of ρ ) equal to 0 means that there is no linear relationship . If we plot the values of the two variables on the x and y axis in a so- called scatterplot (or scatter diagram) then they form a "cloud" around a horizontal line. Ava lueof r close to 1or -1 means that the dots in the scatterplot lie close to a straight line and it implies a strong (linear) correlation. Ava lueo f r
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The sample correlation coe cient between two variables X and Y is calculated as r = P ( x i ¯ x )( y i ¯ y ) q P ( x i ¯ x ) 2 q P ( y i ¯ y ) 2 = P x i y i P x i P y i n r P x 2 i ( P x i ) 2 n r P y 2 i ( P y i ) 2 n = SS xy SS xx q SS yy where SS stands for sum of squares . To test the null hypothesis H 0 : ρ =0 we may use the following test statistic t = r s n 2 1 r 2 which under the null is distributed according to the t distribution with n 2 degrees of freedom.
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Example 13-7: Test whether the linear correlation between income and food expenditure is positive at the 1% signi f cance level. H
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lecturenotes09 - ECON 41 Statistics for Economists Lecture...

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