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I am running a regression analysis between S&P 500's rate of change and Vanguard's Energy ETF rate of change. I would like to know whether the assumptions of linear regression are fulfilled.

Referring to heteroscedasticity, I created the plot between standardized residuals and standardized predictive values. Can I extract something from the following plot? enter image description here

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You need to be cautious here, because you are dealing with time series data.

If this was a cross-sectional dataset, you could simply use the Breusch-Pagan test to screen for heteroscedasticity at the 5% significance level. Note that all code in this example is denoted in R.

bptest(y~x)

The image of the residuals themselves do not show the classic "funnel-shape" one would expect to see if heteroscedasticity were present.

However, you need to make sure that other elements of your model also satisfy the assumption of BLUE (Best Linear Unbiased Estimator).

Therefore, if you have not already, you should make sure that you have tested your data for autocorrelation. This can be done by using the Durbin-Watson test.

dwtest(y~x)

If serial correlation is detected and you need to conduct any transformations on your model, e.g. first differencing, then run the test for heteroscedasticity again using the updated time series.

With the above having been said, I would recommend investigating your data using an ARCH (Autoregressive Conditional Heteroscedasticity), since this is geared towards detecting heteroscedasticity that is being caused by time series related correlations.

The following document should be insightful:

people.bath.ac.uk/bm232/EC50162/ARCH.doc

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  • $\begingroup$ Amended. Apologies for any confusion. $\endgroup$ May 8, 2017 at 12:03

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