# Deriving Single Index Model (Market Model)

$$R_{it}=\alpha_i+\beta_i\cdot R_{mkt}+\epsilon_{it}$$

• $$R_{it}$$ is the return of the stock of observation
• $$R_{mkt}$$ 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 $$R_{mkt}$$ in relation to this model? (i.e. simple annual mean, annual ewm etc.)

• 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