# If the ACF of a time series is within the 95% bounds, is it white noise?

I have a detrended series where the ACF and PACF has lags all within the 95% confidence bounds. This would suggest the series is a White Noise. However, fitting it to an ARMA model (in R) gives the following output:

Coefficient(s):
Estimate  Std. Error  t value Pr(>|t|)

ar1 ,0.9871, 0.0036, 271.907, < 2e-16

ma1 ,-0.9949, 0.0022, -435.129, < 2e-16

intercept ,0.0024, 0.0006, 3.546, 0.000392

Fit:
sigma^2 estimated as 0.03175,  Conditional Sum-of-Squares = 158.74,  AIC = -3054.03


Thus I would be inclined to think it is ARMA(1,1), not white noise. How do I combine these two pieces of information?

You over-modeled your data as the ar coefficient .9871 is (nearly) cancelled by the ma coefficient .9949 . Your series is probably white noise although I would need your data to confirm this as anomalies may be present masking/confusing model identification.

In response to your query ...

 ARMA (p, q) Models in Lag Operator Form

For an ARMA (p, q) model of order (1, 1), it can be written as:

(1 -phiB )y(t)  =   (1 + thetaB ) a(t)                             …

If phi = -theta for ALL phi

y(t)    =   a(t)


The point simply is that if yt is white noise then there are an infinite number of possible values for phi.

you could have used a lag of 12 for both of your factors and obtained similar self-cancelling coefficients.

I have looked at various refereed articles and have not come up with anything better than this.

• No, I did not use auto.arima. I used ARMA command from tseries library. – student1 Apr 15 '19 at 19:32
• Do you have any reference regarding how ar and ma coefficients cancel each other to produce noise? I didn't see that before. – student1 Apr 15 '19 at 20:27