<p>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.</p>

CC BY-SA 3.0
2016-03-10T17:16:35.970Z
2016-03-10T17:16:35.970Z
Andrew M
<p>I believe that @Andrew_M is right, this is caused by differences in default options in the implementations of ARIMA across statistical applications.</p>

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

<p>Results:</p>

<pre><code>&gt; 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
&gt; 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
</code></pre>

CC BY-SA 3.0
2016-04-14T11:04:21.867Z
2016-04-14T15:17:56.807Z
2016-04-14T15:17:56.807Z
BKay
<p>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.</p>

<p>My results (in R):</p>

<pre><code>&gt; arima010a

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


sigma^2 estimated as 19.23:  log likelihood = -2126.56,  aic = 4255.11
&gt; 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
</code></pre>

<p>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?</p>

<p>I'm looking forward to any answer.</p>

<hr>

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

CC BY-SA 3.0
2016-04-13T22:17:01.757Z
Another possibility: Estimating these models requires numerical optimization, wich does not neccesarily converge to the true solution.
InfiniteVariance
<econometrics><time-series><statistics><applied-econometrics><autoregressive>
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?
Übel Yildmar
# 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?

(Asked by: username_0 on 2016-03-10)

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.

**Tags:** econometrics, time-series, statistics, applied-econometrics, autoregressive
> **Comments:**
> **username_1**: Another possibility: Estimating these models requires numerical optimization, wich does not neccesarily converge to the true solution.  

---

## 2 answers

### Answer by username_2 on 2016-03-10. Score: 2

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.


---

### Answer by username_3 on 2016-04-14. Score: 1

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


---

