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ARIMA Transfer Function Sign Switch?

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Posts: 3

ARIMA Transfer Function Sign Switch?

Hi all,


I am running zero-order, first-order, and pulse transfer functions for (0,1,1) models in SAS 9.4. When I ran the identification and estimation steps the MA(1) parameter came out negative when it should have been positive. I expected that since I read that SAS will reverse the MA sign. 


I have run the three transfer functions and the MA(1), omega, and delta parameters are all reported negative and the mu is reported positive (screen shot attached). Likewise with the t-ratios. It also makes me wonder if the AIC and SBC signs are reversed too. 


Is it safe to assume that SAS 9.4 is reversing the parameter and t-ratios signs? I need to make sure this assumption is correct because diagnostic protocol requires I reject parameters that are not between 0 and 1. 


(Lastly, as a side note, the screen shot attached is of one of the three transfer functions, which you can see the omega and delta are not statisitcaly significant. That is okay since this is one of my rival hypotheses.)

Posts: 88

Re: ARIMA Transfer Function Sign Switch?

The followings are copied from the PROC ARIMA doc. You might see different formulations from other software vendors. We are not aware of any issue with PROC ARIMA having sign switched. thanks


General Notation for ARIMA Models

Notation for Pure ARIMA Models

Mathematically the pure ARIMA model is written as

\[ W_{t}={\mu }+\frac{{\theta }({B})}{{\phi }({B})}a_{t} \]



indexes time


is the response series ${Y_{t}}$ or a difference of the response series

${\mu }$

is the mean term


is the backshift operator; that is, ${{B}X_{t}=X_{t-1}}$

${{\phi }({B})}$

is the autoregressive operator, represented as a polynomial in the backshift operator: ${{\phi }({B})=1-{\phi }_{1}{B}-{\ldots }-{\phi }_{p}{B}^{p}}$

${{\theta }({B})}$

is the moving-average operator, represented as a polynomial in the backshift operator: ${{\theta }({B})=1-{\theta }_{1}{B}-{\ldots }-{\theta }_{q}{B}^{q}}$


is the independent disturbance, also called the random error

The series ${W_{t}}$ is computed by the IDENTIFY statement and is the series processed by the ESTIMATE statement. Thus, ${W_{t}}$ is either the response series Y$_{t}$ or a difference of ${Y_{t}}$ specified by the differencing operators in the IDENTIFY statement.

For simple (nonseasonal) differencing, ${W_{t}=(1-{B})^{d}Y_{t}}$. For seasonal differencing ${W_{t}=(1-{B})^{d}(1-{B}^{s})^{D}Y_{t}}$, where d is the degree of nonseasonal differencing, D is the degree of seasonal differencing, and s is the length of the seasonal cycle.

For example, the mathematical form of the ARIMA(1,1,1) model estimated in the preceding example is

\[ (1-{B})Y_{t}={\mu }+\frac{(1-{\theta }_{1}{B})}{(1-{\phi }_{1}{B})}a_{t} \]
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