Time Series Analysis, Lecture 5: AR and MA Theory

We introduce some of the theory regarding autoregressive and moving average processes. We consider when a process is causal and when it is invertible. We also highlight that for a causal AR process, there exists a non-causal AR process that is stochastically equivalent to it.

[rishikamadan]

thanks for the video! for ease of notation, is it okay to write autocovariance(tau) = E(X_t, X_(t+tau)? the i-j = tau part is kinda confusing....

[adamkolany]

to avoid the whole abracadabra you are making aboat 48:30 with complex numbers, just consider the ring of timeseries (that is the real/complex sequences indexed by nonnegative integers) with Cauchy product as the multiplication. then the "lag operator" is just the sequence <0, 1, 0, 0, 0, 0, …> and all sequences with nonvoid first value is reversible. in particular the reverse of <1, - F, 0, 0, 0, …> is just < 1, F, F^2, F^3, … > without any bullshit about the absolute value of F.

[lekshmip]

Would you please post your references?

[adamkolany]

@56:09 I think you have just completely messed it up. if Fi s the product of these linear parts, then F^-1 is the product of their reciprocals AND you can write it as a sum of fractions whose nominators are these linear pieces. each of them in turn then is an infinite series which you can collect together.

[adamkolany]

@6:54 So as you write that is wrong !! What about the iniitial values of X. That is the values X_0, ..., X_{p-1} ?? You cannot apply the equation for theese values !