Oct 16 2008 LEC 7 ECON 240A 1 Bivariate Relationships L Phillips I Introduction In much of economics we are interested in the relationship between one variable and another For example is the demand for tea sensitive to the price of coffee How sensitive We will look at an example that comes from the capital asset pricing model How much of the variation in the monthly rate of return for the UC stock index fund is explained by variation in the monthly rate of return in the Standard and Poor s Index That is how much does the rate of return on this specific asset the UC stock index fund depend on the market What accounts for the variation that is not explained by the market To answer these questions in this bivariate example we will first rely on exploratory graphical analysis and look at a scatter plot of the data Then we will see how well a linear model fits the data This will introduce us to ordinary least squares and the estimation of the linear returns generating process for an individual security The estimated slope of this linear model is the famous beta an indicator of whether the UC stock index fund is more volatile or less volatile than the market In this example the estimated slope or beta is specific to the UC index fund Since stock index funds are designed to match market behavior the slope should be fairly close to one In exploring the linear model various issues will arise Is the relationship between the dependent variable and the independent variable linear quadratic or some other functional form Often the exploratory graphical analysis provides a clue How well does the linear model fit the data We will develop measures of goodness of fit Using our example of a returns generating process we will estimate beta Oct 16 2008 LEC 7 ECON 240A 2 Bivariate Relationships L Phillips What is the expected value of this estimate What is its variance Is beta significantly different from zero Is beta significantly different from one We will develop hypothesis tests for this slope parameter estimated for the linear model II Capital Asset Pricing Model The data for the monthly rates of return for the UC stock index fund and for the Standard and Poor s Index of 500 stocks is reproduced in Table I along with the rate of return on the 30 day Treasury bill Table I Monthly Rate of Return UC StockIndex Fund S P 500 30 Day Treasury Bill Date August 99 September 99 October 99 November 99 December 99 January 2000 February 2000 March 2000 April 2000 May 2000 June 2000 July 2000 UC Stock Index 2 46 2 44 7 48 3 79 5 48 1 95 2 67 8 78 1 45 0 56 1 97 2 03 S P 500 0 50 2 74 6 23 2 03 8 21 5 02 1 89 9 78 3 01 2 05 2 47 1 56 30 Day Treasury 0 39 0 39 0 39 0 36 0 43 0 43 0 43 0 47 0 46 0 50 0 40 0 48 A scatterplot of the monthly rate of return on the UC stock index fund versus the monthly rate of return on the Standard and Poor s Index of 500 stocks is illustrated in Figure 1 Oct 16 2008 LEC 7 ECON 240A 3 Bivariate Relationships L Phillips Figure 1 Scatterplot of Monthly Rates of Return 10 8 UC Stock Index Fund 6 4 2 0 6 4 2 0 2 4 6 8 10 12 2 4 Standard and Poor s Index of 500 Stocks A plot offers a visual indication of the relationship between two variables whether it is positive or negative linear or nonlinear and whether the relationship is tight or not i e is the goodness of fit high or low In this example note that for ten of the twelve observations the two variables have the same sign Thus there is a strong indication of a relationship with a positive slope There is no visual indication of nonlinearity The data points do not lie along a straight line so the goodness of fit is not perfect A scatterplot often offers a great deal of insight into the relationship between two variables Furthermore exploratory data analysis can be extended to look at the relationship between the dependent variable y and the explanatory variable x Oct 16 2008 LEC 7 ECON 240A 4 Bivariate Relationships L Phillips controlling for a third factor w this can be accomplished in various ways One possibility is to use different symbols for the data points For example in Figure 2 the data points plotted in Figure 1 are differentiated using triangles for 1999 and squares for 2000 Figure 2 Scatterplot of Monthly Rates of Return Triangles 1999 squares 2000 10 8 UC Stock Index Fund 6 4 2 0 6 4 2 0 2 4 6 8 10 12 2 4 Standard and Poor s Index of 500 Stocks There are probably too few data points to distinguish whether the slope or other aspects of the relationship between these two variables has changed between 1999 and 2000 We will return to the topic of graphical multivariate data analysis The specification for the returns generating process for the UC stock index fund as developed in the capital asset pricing model is rUC t rf t rSP t rf t eUC t 1 Oct 16 2008 LEC 7 ECON 240A 5 Bivariate Relationships L Phillips where rUC t is the monthly rate of return on the UC stock index fund and rf t is the risk free rate proxied by the 30 day Treasury Bill rate and presumed known at the beginning of each month The monthly rate of return for the market is measured in this example by rSP t the return for the Standard and Poor s Index of 500 stocks The error term eUC t is a source of variation in the rate of return on the UC stock index fund that is specific to that asset The parameter is the intercept of the linear relationship i e the value of the dependent variable when the explanatory variable is zero assuming an error of zero In equilibrium for the returns generating process the intercept is expected to be zero If the intercept were negative for example the UC stock index fund would be in disequilibrium and the expected return for this asset would be too low as a consequence of being negative and dragging the return down So for example if we were to find an estimated value for the intercept that was significantly below zero we could infer price movement for the UC stock fund We would expect the price to fall raising the monthly rate of return Hence if its intercept is negative this asset would be called overpriced The magnitude of the parameter measures how changes in the market rate of return are translated into changes in the rate of return for the UC stock index fund A significantly greater than one would indicate that the UC stock index …
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