# Estimate of an AR model

I have this part of a project which states this: Once you choose the best three models for each series (according to AIC, the PACF and ACF, and "from general to specific), the next step is to estimate these 6 models. You should present the final models (estimates and t-stats) and the proof that the residuals are white noise (you can use for this Q-stat or check if residuals have serial correlation with 5% significance level). In the previous step I had to find the parameters of the AR model, and in order to check if residuals are white noise I run the Ljung-Box test in this way (given the cutoff of the PACF of my differentiated time series is 6, I wanted to validate an AR(6). I defined x as my diff(log(gdptimeseries)). ar6=arima(x,order=c(6,0,0)

Call:
arima(x = gdplogdiff, order = c(6, 0, 0))

Coefficients:
ar1      ar2     ar3      ar4      ar5     ar6  intercept
3.3058  -4.5471  3.0440  -0.6789  -0.3044  0.1651      0e+00
s.e.  0.0579   0.2025  0.3345   0.3343   0.2021  0.0576      4e-04

sigma^2 estimated as 8.669e-09:  log likelihood = 2250.22,  aic = -4484.44


What I'm I now supposed to do? how do I estimate the parameters and the t-stats of the model?

The estimated values of the autoregressive parameters are already reported in the output you provided:

Coefficients:

ar1      ar2     ar3      ar4      ar5     ar6  intercept
3.3058  -4.5471  3.0440  -0.6789  -0.3044  0.1651      0e+00


The associated standard errors are listed immediately below these estimated values:

         ar1      ar2     ar3      ar4      ar5     ar6  intercept

s.e.  0.0579   0.2025  0.3345   0.3343   0.2021  0.0576      4e-04


The t-values are obtained as the ratio of the estimated autoregressive coefficient to its corresponding standard error. For example, the t-value for testing the null hypothesis thar the (true) first autoregressive coefficient is 0 against the alternative hypothesis that it is different from 0 is given by:

t = 3.3058/0.0579


and you can compare its absolute value against the critical value 1.96 from a standard normal distribution to decide whether you can reject the null hypothesis in favour of the alternative hypothesis based on the evidence in your data at the usual 5% significance level.