# Simple AR(p) question

Hey guys,

Simple question but I’m confused.

Autoregressive model are notated using the AR§ format. Where p = number of lagged values.

So the book frequently using the example of an AR(1) model = b0 + b1 * x(t-1)

They also call an AR§ model with a seaosn lag: AR(1) = b0 + b1 * x(t-1) + b2 * x(t-2)

My question is why isn’t this called an AR(2) model since there are two lagged values in the equation.

Thank you.

you’re lagging the same variable but for different times periods, often used when data has seasonal trends, so it’s still only one variable (AR 1)

But isn’t an AR(2) model by definition, the same variable lagged out to 2 time periods (t-1) and (t-2)?

Autoregressive model are notated using the AR§ format. Where p = number of lagged values the AR model will include as independent variables. AR (2) would be two independent variables. lagging the same variable at different time periods does not create another independent variable.

So it depends on whether you really want to model Xt based on the two previous periods or simply want to make a seasonal adjustment?

So if I had an AR(2) model, and found autocorrelation due to the fact that returns were related to returns 2 months prior, a seasonality adjusted AR(2) model should look like:

AR(2) = b0 + b1 * x(t-1) + b2 * x(t-2) + b3 * x(t-3) + b4 * x(t-4)

Would I be correct in saying that a seasonal adjustment does not change the p-order of magnitude of the AR§ model?

This is correct:

AR(1) = b0 + b1 * x(t-1) AR(2) = b0 + b1 * x(t-1) + b2 * x(t-2) AR(3) = b0 + b1 * x(t-1) + b2 * x(t-2) + b3 * x(t-3) AR(1) with seasonal lag is eg. b0 + b1 * x(t-1) + b2 * x(t-12) As the name implies (auto) we regress the same variable

So if the seasonal month is month 2, how would you write the last equation?

From the link above, it looks like:

AR(4) model y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + b4*y(t-4) + e(t)

AR(4) model with Q4 seasonal lag y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + b4*y(t-4) + b_s*y(t-4) + e(t)

It seems that we add the seasonal lag to whatever AR() model! I’m not sure I understand how adding the same value twice makes sense, but I’ll take that for now.

This model does not make sense.

However it can be estimated using linear regression… but multicilinearity is a huge problem.

T-stats will be undervalued.

So this model is not valid. anymore.

confused…what’s the difference between the two then?

How about AR(12) with season lag? The last 3 will be “b11 * x(t-11) + b12 * x(t-12) + b13 * x(t-12)”?