Confused over AR(2) autoregressive model

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iamsmrt

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Hey all,
I’m confused over the definition of a second-order autoregressive model. I’m on page 229 of Schweser and it says a second order autoregressive model uses two lagged values of the dependent variable. But…. on page 234 when they’re trying to correct for the seasonality, they add a seasonal lag factor…. ie. it’s an equation where the historical autoregression terms are one quarter ago but also 4 quarters ago…. the 4th quarter being the seasonal lag factor.
To me, that sounds like there are two lagged values of the dependent variable, one quarter ago as well as 4 quarters ago. But the book specifically states that this is NOT an AR (2) model, it’s just an AR(1) with a seasonal lag factor. Can someone help me understand this seeming contradiction?
 
Just from the back of my rusty mind…
AR(1) model
y(t) = b0 + b1* y(t-1) + e(t)
AR(2) model
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + e(t)
AR(3) model
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + e(t)
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(p) model
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + b4*y(t-4) + .. bp*y(t-p) + e(t)
Where p = no of lagged values.
So you see, the AR() model equation is PROGRESSIVE… you don’t skip-hop and jump the numbers.
Now the difference in AR(p) and seasonal lag.
AR(1) model with Q4 seasonal lag
y(t) = b0 + b1* y(t-1) + b_s*y(t-4) + e(t)
AR(2) model with Q4 seasonal lag
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b_s*y(t-4) + e(t)
AR(3) model with Q4 seasonal lag
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + b_s*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)
.
.
.
AR(p) model with Q4 seasonal lag
y(t) = b0 + b1* y(t-1) + b2*y(t-2) + b3*y(t-3) + b4*y(t-4) + .. bp*y(t-p) + b_s*y(t-4) + e(t)
Now just try to see if the above equations make sense and you will get the difference. Re-read the 2 pages (I am too lazy to open those Schweser so late in the night) you mentioned in your post and things will be clear.
 
Good explanation…. The cfai could have just written that and save themselves 15 pages of nonsense
 
Thats what i also thought swaption but wasn’t sure…
That SR models are always consecutive; for example a AR(2) model will have two consecutive lag factors: (y-1) and (y-2). but a seasonal AR model will lag factors and a seasonality factor.
 
Having 2 lagged values of the dependent variable =/= AR(2)
For it to be an AR2 model it has to be of the form
Yt = B0 +B1Y(t-1)+B2Y(t-2)+et.
Therefore the model that you mentioned is not an AR(2). Just having 2 of the lagged values of the dependent variable doesnt make it an AR(2) model.
 
It’s like, Factoral thingy, if you like it that way.
What is 5!? 5! = 5*4*3*2*1 (consecutively decreasing till 1 encountered)
Does this make a factoral series == 1*2*4? Nopes! It can be interpreted as (2!)*4
Sorry if it got more confusing. My coffee is just starting to kick in and the market is +50 already. Let me put some limit sell orders ;-)
 
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