# Deriving Single Index Model (Market Model)

$R_{it}&space;=&space;\alpha_i+\beta_i&space;\cdot&space;R_{mt}&space;+&space;\epsilon_{it}$

• $R_{it}$ is the return of the stock of observation
• $R_{mt}$ is the return of the reference market
• $\beta_i$ is the regression coefficient between the observed stock and the reference market
• $\alpha_i$ is the regression intercept between the observed stock and reference
• $\epsilon_{it}$ is the error (a random variable with expectation zero and finite variance)

First of all can someone help me to understand what $\epsilon_{it}$ 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 $R_{mt}$ 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