Forecast biasness of VIX term structure

I'm interested in the topic of VIX futures being overpriced, so I'm looking for different models to find evidence for it. Asensio 2013 uses a regression to evaluate the forecast biasness of the VIX term structure. $$\left(\frac{1}{k}\right)\sum_{i=1}^{k-1}\left[(\sigma^{VIX}_{i,i\rightarrow i+j})^2-(\sigma^{VIX}_{0,0\rightarrow j})^2\right] = \alpha_0 + \beta_0 \left[(\sigma^{VIX}_{0,0\rightarrow k+j})^2-(\sigma^{VIX}_{0,0\rightarrow j})^2\right] + \sum_{i=1}^{k-1}u_i\,.$$ That's equation (14) in the paper, such that

• $u_i$ represent the expectational errors.
• $j$ represents the future date.
• $k$ represents the number of periods.

Now I am asking myself, due to confusion of these indices, which values should I plug into this equation, when I want to apply this to the VIX and VIX futures.

I would be very thankful if someone could explain this to me!

EDIT: First, thanks to Farahvartish for correcting my first version.

Second, Asensio uses in a previous step (equation 13) Campa and Chang's (1995) risk neutral formula to test expectations hypothesis:

$$\sigma^2_{0,km} = (\frac{1}{k})E_0[\sum_{i=0}^{k-1}\sigma^2_{im,(i+1)m}](\frac{\theta_{km}}{\theta_m})$$

• $\theta$ represents the concavity adjustment for a given tenor (assumed to be 1).
• $m$ represents the number of months until expiration.
• $k$ represents the number of periods.

Originally this formula is applied to options. I don't know how to use it for VIX futures and how to imply it to the formula of forecast biasness. Is it that: $$\sigma^2_{0,km} = (\sigma^{VIX}_{0,0\rightarrow j})^2$$