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Economics 173 Business Statistics Lecture 14 Fall, 2001 Professor J. Petry
Hypothesis Test to Evaluate Model: Testing the slope –When no linear relationship exists between two variables, the regression line should be horizontal. Linear relationship. Different inputs (x) yield different outputs (y). No linear relationship. Different inputs (x) yield the same output (y). The slope is not equal to zeroThe slope is equal to zero
We can draw inference about 1 from b 1 by testing H 0 : 1 = 0 H 1 : 1 = 0 (or 0) –The test statistic is –If the error variable is normally distributed, the statistic is Student t distribution with d.f. = n-2. The standard error of b 1. where
Solution –Solving by hand –To compute “t” we need the values of b 1 and s b1. –Using the computer There is overwhelming evidence to infer that the odometer reading affects the auction selling price.
Example –Evaluate the model used in the Armani’s Pizza example by testing the value of the slope. You are given:
Armani Pizza Example -- Output
R 2 (Coefficient of Determination) to Evaluate Model –When we want to measure the strength of the linear relationship, we use the coefficient of determination.
–To understand the significance of this coefficient note: Overall variability in y The regression model Remains, in part, unexplained The error Explained in part by
x1x1 x2x2 y1y1 y2y2 y Two data points (x 1,y 1 ) and (x 2,y 2 ) of a certain sample are shown. Total variation in y = Variation explained by the regression line) + Unexplained variation (error)
R 2 measures the proportion of the variation in y that is explained by the variation in x. Variation in y (SST) = SSR + SSE R 2 takes on any value between zero and one. R 2 = 1: Perfect match between the line and the data points. R 2 = 0: There are no linear relationship between x and y.
Example 17.4 –Find the coefficient of determination for example 17.1; what does this statistic tell you about the model? Solution –Solving by hand; –Using the computer From the regression output we have 65% of the variation in the auction selling price is explained by the variation in odometer reading. The rest (35%) remains unexplained by this model.
Example –Find the coefficient of determination for the Armani’s Pizza example; what does this statistic tell you about the model? You are given values of SSR = , and SSE =
Armani Pizza Example -- Output
Project I: Finance Application Project simulates the job of professional portfolio managers. Your job is to advise high net-worth clients on investment decisions. This often involves “educating” the client. Client is Medical Doctor, with little financial expertise, but serious cash ($1,000,000). The stock market has declined significantly over the last year or so, and she believes now is the time to get in. She gives you five stocks and asks which one she should invest in. How do you respond?
Project I: Finance Application The project is divided into three parts 1.Calculate mean, standard deviation and beta for each asset. –gather 5 years of historical monthly returns for your team’s five companies, the S&P 500 total return index and the 3- month constant maturity Treasury bill. Put in table and explain to client. 2.Illustrate benefits of diversification with concrete example. –You are provided with average annual returns and standard deviations for the S&P 500 and the 3-month T-bill. You create a portfolio with these two instruments weighted differently to illustrate the impact on risk (standard deviation). 3.Analyze impact of correlations on diversification. –Create table of correlation coefficients between all assets, and explain. Select asset pair with r closest to –1. Explain. –Adjust graph in 2 with r = 1, and –1 instead of 0. Explain.
Part 1. Finding Beta (17.6 in text) One of the most important applications of linear regression is the market model. It is assumed that rate of return on a stock (R) is linearly related to the rate of return on the overall market. R = 0 + 1 R m + Rate of return on a particular stockRate of return on some major stock index The beta coefficient measures how sensitive the stock’s rate of return is to changes in the level of the overall market.
Example 17.5 The market model Estimate the market model for Nortel, a stock traded in the Toronto Stock Exchange. Data consisted of monthly percentage return for Nortel and monthly percentage return for all the stocks. This is a measure of the stock’s market related risk. In this sample, for each 1% increase in the TSE return, the average increase in Nortel’s return is.8877%. This is a measure of the total risk embedded in the Nortel stock, that is market-related. Specifically, 31.37% of the variation in Nortel’s return are explained by the variation in the TSE’s returns.
Part 2. Diversification (Ex 6.8, sect 6.7) Investment portfolio diversification –An investor has decided to invest equal amounts of money in two investments. –Find the expected return on the portfolio –If = 1,.5, 0 find the standard deviation of the portfolio.
–The return on the portfolio can be represented by R p = w 1 R 1 + w 2 R 2 =.5R 1 +.5R 2 The relative weights are proportional to the amounts invested. –The variance of the portfolio return is V(R p ) = w 1 2 V(R 1 ) + w 2 2 V(R 1 ) +2w 1 w 2 1 2 –Calculate the portfolio return and risk for = 0, = 1 and = -1
–Substituting the required coefficient of correlation we have: For = 1 : V(R p ) =.1056 =.3250 For =.5: V(R p ) =.0806 =.2839 For = 0: V(R p ) =.0556 =.2358 Larger diversification is expressed by smaller correlation. As the correlation coefficient decreases, the standard deviation decreases too.