# Single-index model

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## Single-index model

A model of stock returns that decomposes influences on returns into a systematic factor, as measured by the return on the broad market index, and firm specific factors. Related: Market Model

## Single-Index Model

The relationship between a security's performance and the performance of a portfolio containing it. The market model states that the security's performance is related to its portfolio's performance, according to its beta. It is calculated as follows:

Return on security = alpha + beta * return on portfolio + residual return
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This is why the shrinkage estimator is a weighted average of the sample covariance matrix with Sharpe's (1963) single-index model estimator where the structure is determined by a shrinkage coefficient k as will be seen in a further section.
the improved covariance matrix, is applied in the paper, S is the sample covariance estimate, I is the covariance matrix derived from a single-index model of stock returns, and k/T [member of][0; 1] is the shrinkage intensity (see Ledoit and Wolf (2003)).
For estimation, we consider the parametric probit model alolng with the semiparametric single-index model estimator of Ichimura (1993).
Single-index models, on the other hand, can accommodate certain forms of heteroskedasticity (general but known form and unknown form if the distribution of the error term depends on x only through the index, i.
Optimization-based estimation methods have been developed for single-index models without making distributional assumptions and thus avoiding misspecification.
Note that we need a location-scale normalization for identification purposes in single-index models.
In single-index models, the asymptotic distribution of the normalized and centered estimator does not depend on the smoothing parameter, so asymptotically, any sequence of smoothing parameters is going to give the same estimate as long as it satisfies certain conditions.
Moreover, comparing these results with those of Table 2, it appears that unconditional alphas from the single-index model are less useful in predicting future performance as measured by excess returns than by their own future values.
By comparing the results from unconditional and conditional alphas, we conclude that in the case of the single-index model it seems that conditional alphas lead to stronger evidence of performance persistence.
Conversely, Lloyd and Shick (1980) determined that the addition of an interest rate index to the single-index model only marginally improved the explanatory power for a sample of commercial bank stock returns.
Past applications of the CAPM and single-index model to forest assets have relied on rates of change in period-average stumpage price (the price of standing timber sold for harvest).
While we focus on the CAPM, the analysis is readily transferred to the single-index model.

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