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About my question: it is a mix between the assumptions of the model and the implementation.

I implemented a DCC(1,1) model for two retrun series (bivariate correlation), with the autoregressive order: 1,1. In total, each return series has 3435 observations (daily returns). For the implemantation I used R (Package ‘rmgarch’).

This are my assumptions for the model:

##Data frame with two return series
Base_Corr <- data.frame(ret.X, ret.Y)

##Specifications for the GARCH model (Volatility part of the DCC)
uspec.Corr = multispec(replicate(2, ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), 
mean.model=list(armaOrder=c(0,0), include.mean=TRUE), distribution.model="norm"))) 
multf.Corr = multifit(uspec.Corr,Base_Corr)

##Specifications for the Correlation (Correlation part of the DCC)
spec.Corr = dccspec(uspec = uspec.Corr, dccOrder = c(1,1), distribution = 'mvnorm')
fit.Corr = dccfit(spec.Corr, data = Base_Corr; fit.control = list(eval.se = TRUE), fit = multf.Corr)

I obtained the follwoing results:

*---------------------------------*
*          DCC GARCH Fit          *
*---------------------------------*

Distribution         :  mvnorm
Model                :  DCC(1,1)
No. Parameters       :  11
[VAR GARCH DCC UncQ] : [0+8+2+1]
No. Series           :  2
No. Obs.             :  3435
Log-Likelihood       :  22738.88
Av.Log-Likelihood    :  6.62 

Optimal Parameters
-----------------------------------
                             Estimate  Std. Error  t value Pr(>|t|)
[ret.US_Equity_REIT].mu      0.000609    0.000163   3.7322 0.000190
[ret.US_Equity_REIT].omega   0.000002    0.000002   1.0786 0.280774
[ret.US_Equity_REIT].alpha1  0.128133    0.032556   3.9358 0.000083
[ret.US_Equity_REIT].beta1   0.869733    0.029627  29.3562 0.000000
[ret.US_MSCI_Large].mu       0.000804    0.000133   6.0662 0.000000
[ret.US_MSCI_Large].omega    0.000003    0.000003   1.0560 0.290975
[ret.US_MSCI_Large].alpha1   0.145551    0.017921   8.1219 0.000000
[ret.US_MSCI_Large].beta1    0.837296    0.024228  34.5594 0.000000
[Joint]dcca1                 0.043839    0.009706   4.5168 0.000006
[Joint]dccb1                 0.943457    0.014202  66.4328 0.000000

Information Criteria
---------------------
                    
Akaike       -13.233
Bayes        -13.213
Shibata      -13.233
Hannan-Quinn -13.226


Elapsed time : 1.571353

My Question is: In the DCC GARCH Fit I get 3435 Oberservations, why I get 3435 obersavations instead of 3434? Because I assume a lag structure of one day DCC(1,1) and used 3435 observations for each return series.

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If your model were a $p$-th order autoregression estimated by conditional least squares, you would "lose" the first $p$ observations, and the number of fitted values would be $n-p$ where $n$ is the sample size. If you used full maximimum likelihood estimation instead, you would not "lose" any observations and you would have as many fitted values as there are observations in your sample.

Your model is not an autoregression, nor is it fit using conditional least squares. The GARCH-DCC model is fit using maximum likelihood, and no observations are "lost", so you have $n$ fitted values. (You would observe a similar behavior if you estimated a simple univariate GARCH model, too.)

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