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Jorge Reyes is a financial analyst with Valores de Playa SA de CV, located in a suburb of Mexico City, Mexico. Two nights a week, he works as an adjunct professor at a local technical institute, lecturing in investments and serving as a consultant in statistics and related fields.During a lecture on modern portfolio theory (MPT), Reyes points out the role that regression plays in estimating the parameters of the capital asset pricing model (CAPM). As an exercise, Reyes presents the results of a regression of returns (Rt) on the company that owns the Mexican stock exchange (ticker symbol BOLSAA.MX) against the U.S. dollar-Mexican peso exchange rate (Et). The data cover the period from late 2011 through early 2012. There are 64 daily observations in the study. Exhibit 1 reports the results of the regression. One of the students asks Reyes about the “Adjusted R2” reported in Exhibit 1. Reyes explains that the adjusted R2 removes the effects of serial correlation in the data.A second student recalls that the presence of heteroskedasticity affects interpretation of the test statistics computed by a regression. Reyes confirms that that is true and suggests the students examine a plot of the predicted BOLSAA return values versus their actual values. Exhibit 2 provides such a graph.Exhibit 2The Data Points and the Fitted Regression Line Explaining theBOLSAA Returns Using the USD-MXNExchange Rate Interpreting the graph, Reyes states:“The presence of heteroskedasticity is indicated when there is a systematic relationship between the values of residuals and the independent variable. It is difficult to see such a systematic relationship in Exhibit 2. Therefore, heteroskedasticity does not appear to be a problem in this regression.”In a later exercise, Reyes asks his students to consider a time series of 622 weekly prices of Maya 22 crude oil. A substantial proportion of Mexico’s oil production is Maya 22 heavy crude. The period of the study is from January 1997 through December 2008.Reyes starts the analysis by looking at a chart of the time series (not shown). Reyes points out several key features of the chart.• First, the prices exhibit an exponential trend in the price increases leading up to 2008.• Second, price behavior in the last few months of 2008 is significantly different from price behavior leading to the market topReyes asks the students to model the time series for the period January 3, 1997, through July 18, 2008, when prices hit the high value of USD126.58. At Reyes’ suggestion, the students first model the prices as an exponential trend (log-linear model). They test for correlated errors from the model using the Durbin-Watson statistic. The results are reported in Exhibit 3. Reyes next suggests they use a first-order autoregressive model (AR(1)). To reduce the impact of the exponential trend, the students continue to use the natural logarithms of the prices, but now they also take the first differences of these logarithms of the prices (xt). They fit a first-order autoregressive model (AR(1)) to the differences of logs. The results of the regression are reported in Exhibit 4. As nonstationarity or heteroskedasticity would negatively impact use of the AR(1) model, Reyes asks the students to test for the presence of each. Results of the unit root test of nonstationarity and of a test for the presence of heteroskedasticity are reported in Exhibit 5. |
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