# Degrees of freedom in regression model on GARCH return process

I am looking for the degrees of freedom in a regression on normalized residuals. The model looks like this:

(1) To adjust the time series, the following GARCH(1,1) model is estimated: $\sigma^2_{it}=\lambda_{0i}+\lambda_{1i}\epsilon^2_{it-1}+\lambda_{2i}\sigma^2_{it-1}$

(2) Returns are then normalized by the volatility estimate from the GARCH model. $a_i$ and $b_i$ are chosen such that the mean of $R_{it}$ is zero and the standard deviation is one: $\bar{R}_{it}=a_i+b_i(1/\hat{\sigma}_{it})R_{it}$

(3) These returns are then regressed on last day return, last day return of the market and 4 dummies for the days of the week:
$\bar{R}_{it} = \gamma_{0i} + \gamma_{1i} \bar{R}_{it-1} + \gamma_{2i}R_{mt-1}+\gamma_{3i}D_{1t}+\gamma_{4i}D_{2t}+\gamma_{5i}D_{3t}+\gamma_{6i}D_{4t}+\epsilon_{it}$

(4) Finally two other dummies are regressed on the residuals from model (3):

$\hat{\epsilon}_{it}=\beta_0+\beta_1Z_{1it}+\beta_2Z_{2it}+u_{it}$

I know there are $557$ return observations in the model. I wonder how to get the degrees of freedom for the $t$-statistic of the model (4) coefficients?

Is it simply $557$ - $3$ (for model 1) - $2$ (for model 2) - $7$ (for model 3) - $3$ (for model 4) $=542$?

Thanks a lot for your help!

• I don't think it is as simple as you suggest in the penultimate line. However, I am not competent enough to provide an answer with a proof. – Richard Hardy Sep 28 '15 at 20:32
• This notation is a bit confusing. You should maybe call the "output" from model 2 something else, like $\tilde{R_{it}}$. That makes it not clear in model 3 whether you are regressing on standardized last day returns or not. Maybe you should clear out this ambiguity. – Gumeo Oct 7 '15 at 7:21
• @jeffrey are there any parameters behind the last day return of the market in model 3? – Gumeo Oct 7 '15 at 8:22
• No there are no parameters behind. It is an observable value like an index return (e.g. S&P 500). – jeffrey Oct 7 '15 at 9:35