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What can we say about an asset which $\beta$ calculated using the CAPM model (regressing the excess returns of the stock vs excess returns of the market) is insignificant?

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Look at the process of estimating your $\beta$ (since if you ask about significance, you have an estimation viewpoint): you try to fit a linear model between your returns $r$ and a factor returns $F$ like

$$r = \beta \cdot F + \epsilon,$$

where $\epsilon$ is your tracking error around the factor (or more accurately around the part of your returns explained by the factor).

Your question about $\beta$ being significantly different from zero, can now be read as a linear regression significance question. They is a common knowledge about this.

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