Time series

[Rahul]

Hi Tutor,

Could you please help me to understand the below mentioned concept in the time series? It would be really helpful if you could explain it in simple words and if possible please provide a small numerical example as well.

Core Reading states(Pg29 - Time Series(2))
In the case of a more general ARMA process we encounter the difficulty that the et cannot be deduced from the t z . For example, in the case of ARMA(1,1) an equation which can be solved iteratively for et as long as some starting value e0 is assumed. For an ARMA (p, q) the list of starting values is e0 to eq . The starting values need to be estimated, which is usually carried out by a recursive technique. First assume they are all equal to zero and estimate the alphai and beta j on that basis, then use standard forecasting techniques on the time-reversed process zn....z1 to obtain predicted values for e0.....e(q-1), a method known as back forecasting. These new values can be used as the starting point for another application of the estimation procedure; this continues until the estimates have converged."

Thank you in advance!

 

[Rahul]

Hi Someone could please help me to understand the above topic?
Thank you in Advance

 

[Hariram Sathasivan]

Hi Rahul, I think its just explaining how to iteratively obtain the e_t, the error term from the assumption of e_0 and backforecasting your z's.
Hope the tutor can confirm though, not sure if I'm missing anything.

 

[Andrew Martin]

Hi Rahul

This is a tricky concept and Core Reading doesn't go into any detail here!

This section is describing parameter estimation using the method of least squares (ie obtaining the parameters that minimise the sum of the squared error terms). However, for an ARMA(p,q) model with q > 0 it is not possible to work out the error terms from the observed time series values. For example, consider an MA(1) process with mean 0:

zt = et + bet-1

Say that we observe z1, z2, ..., zn. Rearranging the above to get a formula for et, we get:

et = zt - bet-1

So, for example:

e1 = z1 - b * e0
e2 = z2 - b * e1
etc

However, we don't know e0 and so we can't work out the errors explicitly.

To address this issue, the Core Reading is describing a technique where we first assume some values for a set of error terms (and, depending on the series, we may also need some time series values also) and use these to derive the implied set of error terms based on this assumption (which in general will be functions of the parameters). For example, in the above MA(1) series we could assume e0 = 0, which means:

e1 = z1
e2 = z2 - b * e1 = z2 - b * z1
e3 = z3 - b * e2 = z3 - b*z2 + b^2 z1
etc

We now have explicit expressions for the errors in terms of the observed values (z1, z2, ..., zn are known) and the parameter b. We can then find the value of b that minimises the sum of these squared errors.

We could just stop here, and use that value of b as our estimate. For a large sample size, this may be a reasonable approach. However, the Core Reading mentions that we could then update our assumptions about our starting value. Consider our MA(1) process above and say we used e0 = 0 to get an initial estimate of b. We can now obtain a new estimate of e0 using this parameter value. There are a few ways to do this. One way is to use the time inverted series as follows:

If zt = et + bet-1 in 'forwards time', we can also write this as zt = at + bat+1 in 'backwards time' for some other white noise series, at. Assuming that a_n+1 = 0 and using our estimated value of b, we can derive the implied values of at as follows:

an = zn - b * an+1 = zn
an-1 = zn-1 - b * an = zn-1 - b zn
...
a1 = z1 - b * a2

We then also have:

z0 = a0 + b*a1.

Assuming that a0 = 0 (its expected value) gives our forecast for z0 as b * a1. Considering our original series (going forwards in time), we have:

z0 = e0 + be(-1)

Assuming that e(-1) = 0 gives e0 = z0 = b* a1 as above. So, we can use that as a new starting point and repeat the above process of determining the implied errors at the other times and obtaining a new estimate of b. We can then repeat this whole process over and over again until it converges.

This would be a pain to do by hand and, although easy enough for a computer to do the actual calculations, there is a decent chunk of code required to set it all up even for an MA(1) process.

Hope this helps!

Andy