Hello everyone.
This Tural Sadigov and today we're going to talk about SARIMA processes.
Objectives is to describe seasonal ARIMA models,
which is also called SARIMA models,
and rewrite the seasonal ARIMA models using backshift and difference operators.
So let's remember ARIMA processes Xt.
If we let Y_t to be Delta^d X_t – remember,
Delta is the difference operator,
1-B where B is the backshift operator,
d is the order of differencing.
So, we take difference of X_t d many times, we obtain Y_t,
and then Y_t becomes an ARMA model,
mixed ARMA model. What does it mean?
It has autoregressive parts right here, the p terms,
and it has a moving average parts,
which are a linear combination of the noises.
Now when Y_t is ARMA,
than X_t becomes ARIMA where d is the order of differencing.
Now we can also rewrite this as a polynomial operators.
For example, phi(B) is autoregressive polynomial and theta(B)
is moving average polynomial and this becomes our ARMA model.
Now, but sometimes it is possible that our data might contain some seasonality,
so the way to think about this is the following.
Let's say we are looking at the sales of refrigerators and if you look
at the sales in August of this year and then August of the last year,
there might be some relationship between those two months.
So there might be some seasonality going on and,
in that case, the observations might repeat itself after every,
let's say, s observations.
In this case, 12 observations.
So, for a time series of the monthly observation,
X_t might depend on annual lags.
For example, X_t might depend on X_{t-12},
which is last August;
X_{t-24}, which was August two years ago; and so forth.
In that case, we say that we have
seasonality and the span of the seasonality or the period is s=12.
Now it is also possible that our data comes as quarterly earnings, for example.
We have looked at such data.
We're going to revisit Johnson and Johnson
which was about quarterly earnings of a company.
In that case, the span of the seasonality is actually just four.
So, in that case,
we will like to discuss seasonal ARIMA models,
and Box and Jenkins basically developed these models.
So, what is a pure seasonal ARMA process?
Well, seasonal ARMA process is basically ARMA process but we have instead of little p, q,
we use capital P and Q for the order of the autoregressive terms,
order of the moving average terms,
and s is for span of the seasonality.
And we have the following format.
Only difference between this equation or
this process from the ARMA is that we have this is now at the s here.
Autoregressive polynomial is the following:
1- Phi B^s – not B – B^{2s} – not B^2 – and so forth.
And moving average polynomial is exactly,
very similar, not exactly the same,
it's very similar, but we have B^s,
B^{2s}, and so forth.
Now, just like in the mixed ARMA process,
we want our process,
seasonal pure seasonal ARMA process or pure SARMA process,
to be stationary and invertible.
And for that reason, just like before,
we're going to require that these polynomials have these complex roots and
all of those complex roots are outside of a unit circle.
So let me give you an example.
For example, if I have ARMA(1,0)_12.
So, moving average. I have P=1.
I'm sorry, autoregressive order P=1;
moving average order is zero,
so I don't have any moving average term;
and seasonality is 12.
We basically have only polynomial of autoregressive polynomial of degree one,
but it's not really degree one, it's degree one times s, which is 12.
And if I rewrite this,
if I expand it and rewrite it,
you can see that X_t here depends on annual lags.
For example, if this is a monthly data,
it depends on X_{t-12} and plus some noise.
Let's look at ARMA(1,1)_12.
In this case, we don't now only have autoregressive polynomial,
we also have moving average polynomial.
Again, degree one times s, which is 12.
And if I expand it,
we obtain that X_t depends on X_{t-12},
but it also depends on the noise from last year if this was a monthly data.
Now, in general, not just pure seasonal ARMA process,
if you look at seasonal ARIMA process,
then we have seven parameters.
We have p, d, q,
capital P, capital D,
captive Q, and s. And this is the polynomial form of that process.
We have (1- B)^d;
this is basically the difference operator d many times.
This is coming from I here, ARIMA is I.
And I also have a seasonal differencing: (1-B^S)^D.
So this is seasonal differencing,
this is non-seasonal differencing.
We have usual autoregressive polynomial,
but we also have
seasonal autoregressive process – I'm sorry – seasonal autoregressive polynomial.
If you look at the right-hand side,
we have moving average polynomial,
we have seasonal moving
average polynomial as well and all of them are specified right here.
In this SARIMA models,
basically we have two parts.
We have a non-seasonal part and we have a seasonal part.
So p here is the order of non-seasonal AR terms,
d is the order of non-seasonal differencing,
q is the order of non-seasonal moving average terms,
capital P is the order of seasonal autoregressive terms.
In other words, sometimes we say SR, right, SAR terms.
D, capital D, is the order of seasonal differencing,
in other words, (1-B^s).
And Q is going to be order of seasonal MA terms and
sometimes we're gonna write this as SMA terms.
Now, as in ARIMA processes,
differencing – we don't have much differencing usually,
it's either one or two in practice.
So if D, the capital D=1,
then Delta operator – this is Delta_s – seasonal differencing X_t is (1-B^s)X_t,
and this is basically X_t-X_{t-s}.
So you look at the differences.
So if this was again monthly data,
and this is basically we are looking at the differences of
the sales in last August and this August.
If D=2, then we are looking at differencing, double differencing, right?
It's gonna be (1-B^s)^2.
If I expand it and I open it up, it becomes X_t-2X_{t-s}+X_{t-2s}.
So let me give you example of SARIMA process.
So this SARIMA model, SARIMA process (1,0,0,1,0,1)_12.
So this is little p, little d, little q.
This is capital P,
capital D and capital Q and seasonality is 12.
I can see that there is no differencing,
so I don't expect any differencing in my model.
And there is no moving average term in my model,
but there is seasonal moving average terms.
There is seasonal autoregressive terms and there's
usual non-seasonal autoregressive terms as well.
So (1-phi(B)) – that is basically coming from this one,
degree of that polynomial is one here.
This is coming from this one which is the degree of seasonal autoregressive polynomial,
which is one times 12,
which is 12 here.
And this part, this is seasonal moving average polynomial with degree one times 12.
If I expand this just like a polynomial and if I expand this,
and I can obtain that X_t is actually depends on X_{t-1}.
So, X_t depends on the previous lag,
it depends on previous year,
and surprisingly, it also depends on X_{t-13}.
Right? So this is one lag before the last year's data.
And it also depends on noise from last year as well.
Let me give you another example.
This is SARIMA(0,1,1,0,0,1).
Here, we do not have autoregressive terms or seasonal autoregressive terms,
and we do not have seasonal differencing,
but we have moving average terms;
we have seasonal moving average terms;
we also have non-seasonal differencing.
Now this four here is the span of the seasonality,
so you can think of this as a quarterly data.
So (1-B), it's coming from non-seasonal differencing; and (1+Theta_1(B)),
that's coming from non-seasonal moving average terms;
and (1+Theta_1 B^4), that comes
from seasonal moving average terms with the degree becomes 1 times 4, which is 4.
If I expand this and put everything to the right-hand side,
we obtain that X_t depends on X_{t-1}.
This is because of non-seasonal differencing.
And then we have noises from previous lags.
Z_t, Z_{t-1}, this is coming from moving average part,
and Z_{t-4}, Z_{t-5}, this is from seasonal moving average part in the model.
So, what have you learned?
You have learned how to describe
seasonal autoregressive integrated, moving average models;
and you have learned how to rewrite seasonal autoregressive moving
average models using backshift and difference operators.