Inconsistent Ljung-Box test result and plot of autocorrelation function of residuals

Question

I get an inconsistent result for the Ljung-Box test: in fact when I run it using the Box.test function it doesn't make me reject the null hypothesis of residuals being white noise, but when I plot the result they're clearly not white noise. The lag used is 72 because I was told to use a quarter of the total number of observations. I'm doing it with AR(6) because from the PACF there's a clear cutoff after lag 6.

> pacf(ts(gdplogdiff),main="PACF of GDP time series")
> acf(ts(gdplogdiff),main="ACF of GDP time series")
> plot(diff(loggpdtimeseries),ylab="Differentiated log")
> ar6=arima(gdplogdiff,order=c(6,0,0))
> Box.test(ar6$residuals,lag=72,type="Ljung-Box",fitdf = 6)
    Box-Ljung test

data:  ar6$residuals
X-squared = 75.512, df = 66, p-value = 0.1981

And this is the plot of the following code

acf(ar6$residuals)

Answer 1

I would suggest reducing the 72 to say 24 or 12 as the Ljung-Box test has not passed the test of time ( that's a joke ! ) as it's assymptotic properties have been called into serious question. See Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey and elsehwere .....

Time to tell your instructor and the authors of your textbook the news !