Correlation between AR[p] processes

[fboehlandt]

Hello everyone,

I would like to compare five timeseries using their correlation coefficient with one another. The five series display significant autocorrelation. The AR[p] processes are unexpected in the context of the type of data (essentially similar to stock prices) and are attributable to the data generating process (self reported prices and performance-smoothing).

Three series are best described by univariate models of the type AR[1], two series by ARMA(1,1) models. The AR/MA processes can be assumed to be unnatural and result from the manipulation of the original data prior to reporting. Hence, the correlation between the timeseries may be over-/understated. The series should be manipulated so as to remove the AR/MA processes from the reported series.

My idea was to specify the best fitting ARMA models for the above series (in terms of Information criterion) and then use the residuals of those regressions to determine the factual correlation not attributable to autocorrelation and disturbance terms. Perhaps someone has come across a similar problem? Thanks in advance

[trubador]

OK, so if the autoregressive behavior is not (or should not be) part of data generation process, then what the actual dynamics of a given series will look like? Do you expect to observe a random walk or pure stochastic behavior? What about stationarity and heteroscedasticity?

[fboehlandt]

I would expect random walk behaviour. The series are return observations, non-stationarity is not really a concern. The effect from return-smooting gradually dies away (usually around t + 3). Testing for heteroskedasticity is inconclusive.

[trubador]

As far as I understand, the observed series are combination of two unobserved parts one of which is the unwanted autoregressive component. However, unless you control the "random walk part", ARIMA modeling will be misleading. You can try the state space framework, but there is no guarantee that it will work smoothly. The code below demonstrates both approaches and compare the results. You'll see that decomposition yields less efficient but unbiased results:

Code: Select all

wfcreate u 200 series x series y series z !nsim = 1000 'number of simulations matrix(!nsim,2) resmat equation ar1 sspace ssm 'True AR(1) coefficent !b = 0.6 smpl @first @first x = 0 y = 0 z = 0 'Decomposition model within state space framework ssm.append @signal z = xs + ys ssm.append @state xs = c(1)*xs(-1) + [var = (c(2))^2] ssm.append @state ys = ys(-1) + [var = (c(3))^2] mode quiet for !i=1 to !nsim 'Hypothetical series smpl @first+1 @last x = !b*x(-1) + 4*nrnd 'autoregressive part y = y(-1) + 2*nrnd '(f)actual component z = x + y 'observed series 'Estimate with AR(1) c=0 ar1.ls z c ar(1) resmat(!i,1) = ar1.@coefs(2) 'Estimate with State Space c=0.1 ssm.ml(c=1e-05) resmat(!i,2) = c(1) 'You can follow the process from the status line %status = "Iteration "+@str(!i)+" of "+@str(!nsim) statusline %status next resmat.distplot(s) hist theory