I'm going to build a multi-factor APT model for the Swiss market starting from the work made by Chen, Roll and Ross (to which I will add and test some additional factors). I have some doubts though:

  1. The APT prescribes that each factor should be expressed as its unexpected component:$$\tilde{F_t}=F_{t}-E[F_t \mid t-1]$$ What if it is not the case? Are results highly affected? I was thinking to put for example consumer confidence, changes in oil or electricity prices, etc. for which it is difficult to find forecasts or for which it is not that cleaver to assume that individuals use simple forecasting techniques like the naïve or drift method. Chen, Roll and Ross use for example changes in monthly industrial production, which is not expressed as its unexpected component.
  2. Are 10 years of monthly observations enough to make a good estimate or should I use more/less observations?
  3. My idea was to apply my model on every stock that is actually registered on the SIX Swiss Exchange. Is it possible? I've read that generally this models are applied to portfolios of stocks. Does this make a huge difference?
  4. Once the estimates are done my idea was to build $K$ factors portfolios (with $K$ being the number of factors). The procedure I had in mind was to solve a quadratic programming problem of the type: $$\min_w w'\Sigma w$$ $$s.t. \ B w \leq b$$ where: $w$ is a vector of weights, $\Sigma$ is the variance-covariance matrix, $B$ is a matrix of $K+1$ rows (where the $K+1$th row is filled with ones) and $N$ columns (with $N$ being the number of stocks analyzed), $b$ is a vector of constraints and $\beta_{k, j}$ is the sensitivity of stock $j$ to factor $k$. As an example: to find weights of the portfolio which mimics the 2nd factor (with $K=3$ and $N=4$): $$B =\begin{bmatrix} \beta_{1,1} & \beta_{1,2} & \beta_{1,3} & \beta_{1,4}\\ \beta_{2,1} & \beta_{2,2} &\beta_{2,3} & \beta_{2,4}\\ \beta_{3,1} & \beta_{3,2}& \beta_{3,3} & \beta_{3,4}\\ 1 & 1 & 1 & 1 \end{bmatrix} \qquad b=\begin{bmatrix} 0\\ 1\\ 0\\ 1\\ \end{bmatrix}$$ Is this the right procedure?

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