# Deriving Single Index Model (Market Model)

• is the return of the stock of observation
• is the return of the reference market
• is the regression coefficient between the observed stock and the reference market
• is the regression intercept between the observed stock and reference
• is the error (a random variable with expectation zero and finite variance)

First of all can someone help me to understand what represents in the model and how one should compute it (considering it is a random variable which distribution is often used?, why expectation zero?)

Secondly from the empirical point of view which is the best practice in computing in relation to this model? (simple annual mean, annual ewm etc of its time series/historical data?)

• The residuals, $\epsilon _{it}$, follow a Normal distribution with zero mean and some variance. That's the definition of noise (if it wasn't noise and you knew it, you could and should model it). As per $R_{mt}$, there's no best practice: if you have developed a good model, either based on econometrics, machine learning, personal experience or dark magic, you should be already working for some of the biggest funds out there. – Lisa Ann Jun 2 '19 at 12:54
• $R_{mt}$ is the return on portfolio broad enough to represent "the market" as a whole. Numerous indices of market performance exist (such as S&P500) and are used for this purpose. The most common practice is to measure these returns on a monthly (or sometimes daily) basis. (It is important not to use any averaging or smoothing, but to use the raw returns). – Alex C Jun 2 '19 at 13:54
• @Alex C: I thought he was talking about forecasts rather than parameters estimation. – Lisa Ann Jun 2 '19 at 16:27
• Good point. The whole question is not as clear as it could be. – Alex C Jun 2 '19 at 16:30
• @AlexC thank you for your reply too. What I would like to understand is considering a given index (of course it should be representative of the stock observed etc.) how one should computes Rmt in order to obtain with this model a consistent and good result? I mean there is no sense in implementing a more complex model for estimating return of a stock (more complex compared to i.e estimating stock return simply with annualized average of daily returns) using "low quality" inputs (i.e computing Rmt as simple annualized average of daily returns - thus limiting the model itself). – Nipper Jun 2 '19 at 17:04