Whitening a regression with an AR process I was reading a research paper:
$Y_{t}\text{=}\beta_{0}+\beta_{1}X_{1t}+\beta_{2}X_{2t}$
(where $Y_{t}$ is stock returns and not the change in stock returns)
($X{}_{1t}$ is the return of a stock market index and $X_{2t}$ is the return of a bond index)
They think $X_{2t}$ follows an AR(3) process and they mention two things in the paper that I want to inquire about:
1- They mentioning that the results with and without whitening are similar
2- They mention that the interest rate series is the residual of the AR(3) process.
My question is the following : how did they whiten the process?
 A: I expect you are talking abour the article: “The Effect of Interest Rate Changes on the Common Stock Returns of Financial Institutions”. I further expect that you have some working knowledge of ARIMA estimation and model fitting. If not, please let me know and I will elaborate on my answer.
We have equation (2) in their article which they want to estimate: ${Y_{t}\text{=}\beta_{0}+\beta_{1}X_{1t}+\beta_{2}X_{2t}+\varepsilon_{t}}
 $ where $Y_{t}
 $, $X_{1t}
 $ and $X_{2t}
 $ are interest rate indices. They say on p. 1145: “If these three interest rates indices measure unanticipated changes in the interest rates, then the series should be white noise processes.” They test this by looking at the ACF, PACF and the Ljung-Box Q statistics of each of these three series and find that there is autocorrelation in the series, hence they are not white noise which is neccessary if these series measure unanticipated changes in the interest rate. They then write on p. 1146: “To obtain a measure of unanticipated interest rate changes, a third-order autoregressive model AR(3) was estimated for each series." Here they fit an AR(3) model for each of thes series. Note that if the series are adequately represented by the AR(3) the residual of this series should be white noise! They essentially estimate these three equations and save the fitted residuals:
$Y_{t}=\delta+\theta_{1}Y_{t-1}+\theta_{2}Y_{t-2}+\theta_{3}Y_{t-3}+u_{t};\; u_{t}\sim N\left(0,\,\sigma^{2}\right)
 $
$X_{1t}=\delta+\theta_{1}X_{1t-1}+\theta_{2}X_{1t-2}+\theta_{3}X_{1t-3}+v_{t};\; v_{t}\sim N\left(0,\,\sigma^{2}\right)
 $
$X_{2t}=\delta+\theta_{1}X_{2t-1}+\theta_{2}X_{2t-2}+\theta_{3}X_{2t-3}+w_{t};\; w_{t}\sim N\left(0,\,\sigma^{2}\right)
 $
What this means is that all the saved residuals from the fitted series are white noise. The then use the residuals of these three series as the interest rate indices as they are white noise and hence: “If these three interest rates indices measure unanticipated changes in the interest rates, then the series should be white noise processes.” (p. 1145). Essentially they estimate equation (2) with $Y_{t}=u_{t}
 $, $X_{1t}=v_{t}
 $ and $X_{2t}=w_{t}
 $, rather then to use the origianl interest indices they use the residuals from the AR(3) model they fitted for each series. This is how they make the series white noise or “whiten” it as they say.
What they mention in footnote 10 is more like a robustness test: “Equation (2) was also estimated without whitening the interest rate series. The results obtained are virtually identical to the results reported in Sections III and IV.” What they do here is to estimate equation (2) with the original interest indices!
The whitened regression: $u_{t}=\beta_{0}+\beta_{1}v_{t}+\beta_{2}w_{t}+error_{t}
 $
The un-whitened regression $Y_{t}\text{=}\beta_{0}+\beta_{1}X_{1t}+\beta_{2}X_{2t}+\varepsilon_{t}
 $
