EXERCISES 3. S&P; 500 Index Prices. Data for the S&P; 500 Stock Index for the ti

Question

EXERCISES 3. S&P; 500 Index Prices. Data for the S&P; 500 Stock Index for the time period January 1974 through December 2002 are in the file named SP5006 on the CD. These data were obtained from the web site www.economagic.com. The objective is to build a regression model relating the current price to the price in the previous month. The regression equation can be written as where represents the S&P; 500 index price in time period i Figure 6.29 is the output for the regression of the current S&P; 500 index price on the previous month's price. Figure 6.30 is the plot of the stan- dardized residuals versus the fitted values. 6.6 Assessing the Assumption That the Disturtances Are Nermally Distrituted 229 FIGURE 6.29 Regression of Current SLP S00 Index Price on Previous Month's Price. Variable Stat P Value ntercept061 SPLAG 2o 0.159 0.000 1.41 0.998917 0.003629 Standard Er-85.460-q-95 R-salad)-99.5 Analysis of Variance sum of Spares mean Square ,Stat PValue 1553606990 345 2520622 46 556127613 75772.720.000 ErrOr Total 306 FIGURE 6.30 Plot of Scatterplot of SRES1 versus FITS1 5.0 Residuals versus Fitted Values for S&P; 500 Index Exercise. 2.5 0.0 -5.0 1000 3000 4000 5000 FITS1 Is there evidence that the constant variance as- sumption has been violated? Justify your answer. Suggest a corection for the violation of the con- stant variance assumption for this example. Try the correction using a regression routine. Does the correction appear to have eliminated the problem of nonconstant variance? State why or why mot.

Explanation / Answer

Lets first discuss what can be inferred from the standardized residual vs predicted values plot.

A scatterplot of standardized residuals vs predicted values indicate:

1. linear assumption is right between the regressor and regressed if the scatterplot is scattered without any definite pattern around zero.

2. The erro variance are all equal (constant) if the rsiduals form a band around 0

3. No particular error point stand out indicating an outlier.

In the graph given above the errors are not surrounded around 0 in a haphazard manner. Instead there is a divergence trend as the fitted values increase. This indicates that the assumption of constant error variance is not valid.

There are 2 major corrections possible:

1. This is an autoregressive data, i.e y(i) is used to predict y(i+1). Hence there is auto-correlation within the errors. So error variance is not constant.

Instead of modelling y(i), we can model del(y(i)) = y(i) - y(i-1). Ususlly the autocovariance moves away on taking the first order difference and we can regress del(y(i)) on del(y(i-1)) and the errors will be uncorrelated and have constant variance.

2. The second solution is: when the scatter plot of residuals vs fittted values diverge as fitted values increase this often indicate non-linearity of the regressor and regressed variables. It generally indicates a quadratic relation. So a good idea will be to regress y(i) on y(i-1)2.

<You have not provided the data for me to check, but I trust both the above solutions will suffice>

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