# 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?

## 1 Answer

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.

ARIMA model identification should follow the following paradigm https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf

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