Bakshi et al., (2006) Estimation of continuous-time models with an application to equity volatility dynamics (Table 2) estimate the following Cox-Ingersoll-Ross model for market variance, $\sigma^2_t$:
$\mathrm{d}\sigma^2_t = (\alpha_0 + \alpha_1\sigma^2_t)\mathrm{d}t + \sqrt{\beta_1}\sigma_t\mathrm{d}W_t$
To estimate their model they use $\left(\frac{VIX_t}{100}\right)^2$ as a proxy to $\sigma^2_t$, where $VIX_t$ is the daily VIX price.
But VIX measures expected volatility (in percentage terms) of the market over the next 30-day period (as implied by S&P index options). So $\left(\frac{VIX_t}{100}\right)^2$ is basically a moving average over future daily market variance. This extra MA structure makes it a poor proxy to the true instantaneous market variance $\sigma^2_t$---especially when trying to model daily market variance dynamics.
What am I missing? Have I misunderstood something? Or have I understood things correctly and using the $VIX_t$ proxy is considered a "good enough" approach?
NOTE: The authors do end up estimating $(\alpha_0, \alpha_1, \beta_1) = (0.3141, -8.0369, 0.1827)$, which implies a long-run market volatility of 0.20 in annualized terms. That more or less agrees with observations. So maybe "good enough"?