# How to forecast ARMA(0,0)

Hi everyone,

When i'm doing auto.arima on my yearly canada interest rate serie, the best model in (0,0,0) with mean = 0

So, ARMA(0,0) has white noise hypothesis OK but how can i forecast it because my model is "empty" for R ?

EDIT 27/03 : I found a best model than ARMA(0,0) but didn't know if it's OK

Thanks,
Paul

The best forecast would be simply the mean of the data (which is certainly not zero).

It is also very unlikely that the best model is (0,0,0).

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You are encouraged to show us the code that you used to get these results. They are very unlikely as previously mentioned. Maybe you made a mistake?

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So, i will try to explain with my weak english level ^^'

``````dCanada = diff(Canada,1)
``````

I diff my serie then i made PACF : i found a weak autocorrelation at order 6 ``````EACF(dCanada)

AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 o o o o o o o o o o o  o  o  o
1 x o o o o o o o o o o  o  o  o
2 x x o o o o o o o o o  o  o  o
3 x o o o o o o o o o o  o  o  o
4 x o o x o o o o o o o  o  o  o
5 o o o x o x o o o o o  o  o  o
6 o x o o o o o o o o o  o  o  o
7 x x o o o o o o o o o  o  o  o

Box-Ljung test

X-squared = 10.216, df = 12, p-value = 0.597
``````

**So , i made EACF and found a ARMA(0,0) .... really weird... **
Moreover, LB test at lag 12 have p-value > 0.05 ......

``````auto.arima(dCanada, trace=TRUE, ic=c("aic"))

ARIMA(2,0,2) with non-zero mean : Inf
ARIMA(0,0,0) with non-zero mean : 175.2989
ARIMA(1,0,0) with non-zero mean : 175.3013
ARIMA(0,0,1) with non-zero mean : 174.2514
ARIMA(0,0,0) with zero mean     : 173.3635
ARIMA(1,0,1) with non-zero mean : 175.4723

Best model: ARIMA(0,0,0) with zero mean

ARIMA(0,0,0) with zero mean

sigma^2 estimated as 0.7855:  log likelihood=-85.68
AIC=173.36   AICc=173.43   BIC=175.55
``````

So, i decide to make an auto.arima et found ARIMA(0,0,0) with non mean as best model....

But i wasn't convinced by this model because i founded autocorrelation with PACF...
So i test a lot of model and found ARMA(2,2) with a better AIC than the ARMA(0,0) ...

``````ARMA0_0 = Arima(dCanada, order = c(0,0,0), include.mean=FALSE)
ARMA2_2 = Arima(dCanada, order = c(2,0,2), include.mean=FALSE)
coeftest(ARMA2_2)
AIC(ARMA2_2)
AIC(ARMA0_0)

z test of coefficients:

Estimate Std. Error  z value  Pr(>|z|)
ar1 -1.460105   0.114566 -12.7447 < 2.2e-16 ***
ar2 -0.493069   0.113722  -4.3357 1.453e-05 ***
ma1  1.944239   0.068811  28.2549 < 2.2e-16 ***
ma2  0.999974   0.069793  14.3278 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

 166.2296
 173.3635
``````

My forecast seems to be very good with the ARMA(2,2) and all white noise hypothesis IS OK with it.

How can i be sure ARMA(2,2) is the best model ? Why auto.arima don't give me the ARIMA(2,0,2) model directly ?
Can i found better with an AR(6) ? Or MA(6) ?

Thanks,
Paul

Since you have first-differenced the interest rate, your model for the level of the interest rate is ARIMA(0,1,0). In that case the best forecast for the next interest rate is the current interest rate. (If you have a non-zero mean, then that also has to be taken into account.)

Of corse, i'm actually trying to find the best model on my differenced serie to made forcast on it and than on my normal DS serie.

But i find autocorrelation in residuals so i don't think it's ARIMA(0,0,0) like recommanded in auto.arima...

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