# What is the reason why ARIMA(0,1,0) on $y_t$ and ARIMA(0,0,0) on diff($y_t$) are not identical time-series models?

I studied at BA level, that ARIMA(0,1,0) on $y_t$ and ARIMA(0,0,0) on diff($y_t$) are the same models. I am doing the Box–Jenkins model estimation on the historic data of US unemployment rate.

My results (in R):

> arima010a

Call:
arima(x = munrate, order = c(0, 1, 0))

sigma^2 estimated as 19.23:  log likelihood = -2126.56,  aic = 4255.11
> arima010b

Call:
arima(x = diff(munrate), order = c(0, 0, 0))

Coefficients:
intercept
0.0708
s.e.     0.1618

sigma^2 estimated as 19.23:  log likelihood = -2126.46,  aic = 4256.92


As you can see, the information criterion (AIC) differs and the $log(L)$, too. My question is, why do they differ in the two model?

I'm looking forward to any answer.

I tried the same later in gretl, and there wasn't any difference between the two models.

• Another possibility: Estimating these models requires numerical optimization, wich does not neccesarily converge to the true solution. – InfiniteVariance Apr 13 '16 at 22:17

I don't know R code but are you estimating an intercept in the ARIMA(0,1,0) model? Because if not, this could be why there is a difference, since you are estimating an intercept in the ARIMA(0,0,0) model.

I believe that @Andrew_M is right, this is caused by differences in default options in the implementations of ARIMA across statistical applications.

library(forecast)
births <- scan("http://robjhyndman.com/tsdldata/data/nybirths.dat")
birthstimeseries <- ts(births, frequency=12, start=c(1946,1))
level010 <- arima(births, order = c(0,1,0), include.mean=TRUE)
diff000 <- arima(diff(births), order = c(0,0,0), include.mean=FALSE)
print(level010)
print(diff000)


Results:

> print(level010)
Call:
arima(x = births, order = c(0, 1, 0), include.mean = TRUE)

sigma^2 estimated as 2.266:  log likelihood = -305.25,  aic = 612.51
> print(diff000)

Call:
arima(x = diff(births), order = c(0, 0, 0), include.mean = FALSE)

sigma^2 estimated as 2.266:  log likelihood = -305.25,  aic = 612.51