ARIMA vs MA with lagged dependent variable

[karolina_konopczak]

My question concerns ARIMA models. Up till now I estimated them In eViews using ar, ma, sar and sma commands. By chance I’ve noticed that there is a substantial difference between estimation results for such a specification and a specification with lagged dependent variable. Can this difference stem form optimization algorithms’ parameters in both specifications? I would be grateful for some help in this respect.

Thank you in advance,

Karolina

[EViews Gareth]

I'm not sure I understand the question.

Estimating a model with an AR, MA, SAR and SMA term, is not the same thing as estimating a model with a lagged dependent variable term. You should not expect the results to be the same.

[startz]

I'm not sure I understand the question.

Estimating a model with an AR, MA, SAR and SMA term, is not the same thing as estimating a model with a lagged dependent variable term. You should not expect the results to be the same.

This is sometimes confusing because, while what Gareth has says is true, there is a special case in which the two are equivalent.

Code: Select all

ls y y(-1) ls y ar(1)

The difference comes about when there are RHS variables.

[EViews Gareth]

True, but even that special case disappears if you add in MA terms etc...

[startz]

True, but even that special case disappears if you add in MA terms etc...

That's true for

...

but

Code: Select all

ls y y(-1) ma(1) ls y ar(1) ma(1)

are the same

[EViews Gareth]

True, but that's just being pedantic!

[karolina_konopczak]

I'm not sure I understand the question.

Estimating a model with an AR, MA, SAR and SMA term, is not the same thing as estimating a model with a lagged dependent variable term. You should not expect the results to be the same.

Could you, please, explain why ? Aren't these two specifications the same up to the constant term? If so, why do I get completely different forecasts from these equations?

[EViews Gareth]

I'm not sure I understand the question.

Estimating a model with an AR, MA, SAR and SMA term, is not the same thing as estimating a model with a lagged dependent variable term. You should not expect the results to be the same.

Could you, please, explain why ? Aren't these two specifications the same up to the constant term? If so, why do I get completely different forecasts from these equations?

Which two specifications are you talking about?

As Startz pointed out, estimating:

Y AR(1) MA(1)

and estimating

Y Y(-1) MA(1)


Are the same thing. Everything else is pretty much different.

[karolina_konopczak]

Which two specifications are you talking about?

As Startz pointed out, estimating:

Y AR(1) MA(1)

and estimating

Y Y(-1) MA(1)


Are the same thing. Everything else is pretty much different.

I'm talking about any arma vs lagged dependent variable model with ma terms (except for this special case).

[startz]

Which two specifications are you talking about?

As Startz pointed out, estimating:

Y AR(1) MA(1)

and estimating

Y Y(-1) MA(1)


Are the same thing. Everything else is pretty much different.

I'm talking about any arma vs lagged dependent variable model with ma terms (except for this special case).

There's some confusion here. Any pure ARMA(p,q) model is the same in EViews whether specified with AR terms or lags of the dependent variable. (There can be very small coefficient differences due to numerical issues.) Perhaps you should post more specifically what you're trying.

[zinco]

I think I understood the problem proposed by Karolina:

If you run in EViews a simple AR(1) regression using the lagged variable “y c y(-1)” or the AR(1) term “y c ar(1)” you will obtain the same coefficient for the lagged variable y(-1) or the ar(1) term, but the coefficient of the constant c will differ, spite of the two equations represent the same thing.

This happens because Eviews estimate the ARIMA coefficients according the explained by Gujarati’s book Basic Econometrics, Chapter 22, (ok, that is not the best nook about time serires…). For example a AR(1) equation:

(Yt - Ϋ) = β (Yt-1 - Ϋ), where Ϋ is the mean of Y, and I eliminate de Ut (white noise term to simplify the equation)

That is, in an autoregressive model value of Y at time t is expressed as deviations from their mean value. Remember that Y is a stationary process and its mean don’t change.

Let’s work in this equation:

(Yt - Ϋ) = β (Yt-1 - Ϋ)
Yt = β (Yt-1 - Ϋ) + Ϋ
Yt = β Yt-1 - β Ϋ + Ϋ
Yt = (1 - β) Ϋ + β Yt-1

This means that to obtain the constant term in EVeiws you should compute the following: C = (1 - β) Ϋ, where β is the AR(1)’s coefficient and Ϋ is the constant estimated by the model.

Now I do an example with the database “arma1” available in EViews help files:

If the equation is entered: “r c ar(1)”. The coefficient 7.897 is estimated for c; and the coefficient 0.993 is estimated for ar(1). What conduct to the wrong model:

Rt = 7.897 + 0.993 R t-1

But according with the equation above, the right model is:

Rt = (1-0.993) * 7.897 + 0.993 R t-1
Rt = 0.0576 + 0.993 R t-1

To prove this, the following equation is estimated: “r c r(-1)”. The coefficient 0.0575 is estimated for c; and the coefficient 0.993 is estimated for r(-1).

Note that the same coefficient is obtained for ar(1) estimated before (0.993); and the coefficient estimated is the same obtained by the formula “(1 - β) Ϋ” (0.0575).

I hope help you.
If you have any doubt with my english I try to rewrite.