What is this called?

Question

We have several time series: $Y, X_1, X_2, X_3, ..., X_n$

The steps taken are:

1. Regress $X_2, X_3, ..., X_n$ on $X_1$ to get residuals of each $X_{(>1)}$
2. Regress $Y$ on $X_1, r_{X_2}, r_{X_3}, ..., r_{X_n}$ to obtain:
   
   

$$ Y = \alpha + (\beta_{X_1} * X_1) + (\beta_{X_2} * r_{X_2}) + (\beta_{X_3} * r_{X_3}) + ... + (\beta_{X_n} * r_{X_n}) $$

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The question proposed to me was:

The 2nd step is quite straight forward if you pass in time series for the 1st stage. However, if we pass in a variance-covariance matrix in the 1st step, how would you proceed to solve 2nd step?

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I dont know how to begin approaching this question because I am not sure what this technique, (regressing against one factor and the residuals from step 1) is called.

Can anyone answer the question proposed to me, or at least tell me what this is called so I can desperately attempt to figure it out myself?

The concept behind this is that $Y$ is a portfolio of securities that I am attempting to analyze the risk of. The end equation represents a risk function where $X_1$ represents the returns of the overall stock market and $X_{2...n}$ represents the returns of sectors within the market (ex. Technology, Utilities, Consumer Discretionary ... etc.)

Answer 1

assume you have just three time series $Y,X_1,X_2$ then the model is \begin{align} Y&=\beta_1X_1+\beta_2 (X_2-a_1X_1)+\beta_3(X_3-a_2X_1)\\ &=X_1(\beta_1-\beta_2a_1-\beta_3a_2)+\beta_2X_2+\beta_3X_3\\ &=X_1\tilde\beta_1+\beta_2X_2+\beta_3X_3 \end{align} That is not more that a linear model (for simplicity I assumed variables are centered).

If I understand your model well, that is a simple linear model that needs just one step to estimate parameters. Then proposing a two step optimization procedure is not necessary.