Fitting an ARIMA process in INLA

[Nafis Sadat]

Hello!

I'm trying to see how feasible it would be to fit a full ARIMA process in INLA. In the econometrics literature, they define an MA process as lags of the errors, and so I thought if I could use INLA's AR(1) model to analogously fit that.

Here's a snippet of code where I simulated an AR(1) and an ARIMA(1,0,1):

  require(INLA)

  require(data.table)  

  ## AR and MA params

  rho = 0.8

  phi = 0.3

  prec = 10

  n = 1000

  ## Simulate ARIMA(1,0,1)

  dt1 = as.vector(arima.sim(list(order = c(1,0,1), ar = rho, ma = phi), n = n, 

                 sd = sqrt(1/prec)))

  ## Simulate ARIMA(1,0,0)

  dt2 = as.vector(arima.sim(list(order = c(1,0,0), ar = rho), n = n, 

                  sd = sqrt(1/prec)))

  

  ## Make sure we can recover params by fitting the ARIMAs

  arima(dt1, c(1,0,1))  

  arima(dt2, c(1,0,0))

Here's the results from first fitting an AR(1) process with the second dataset. I fit the AR(1) using both the 'ar1' model and also by using the lag of the variable. 

  ## Fit the AR(1) from dt2 in INLA

  ar1_dt2 <- data.table(y = dt2, t = 1:n)

  ar1_dt2[, lag_y := shift(y)]

  

  ## Fit with model

    inla_ar1_dt2_1 <- inla(formula = y ~ f(t, model = 'ar1'),

                         data = ar1_dt2)

    summary(inla_ar1_dt2_1)  

    

  ## Fit as lag FE

    inla_ar1_dt2_2 <- inla(formula = y ~ lag_y,

                           data = ar1_dt2)

    

    summary(inla_ar1_dt2_2)

### Both the models recovered the AR(1) parameter in either the FE or the RE ###  

The results are quite similar and this is where I got a bit confused, because an AR(1) on residuals versus an AR(1) on the dependent variable aren't equivalent right (or maybe I'm wrong..)?

Finally, I attempt to fit an ARIMA(1,0,1) by including the lag value and the 'ar1' model (with the hope that the former will pick up the AR(1) part, and the latter will pick up the MA(1) part): 

  ## What if we try to fit the ARIMA in INLA?

  arima_dt1  = data.table(y = dt1, t = 1:n)

  arima_dt1[, lag_y := shift(y)]

  inla_arima_dt1 <- inla(formula = y ~ lag_y + f(t, model = 'ar1'),

                       data = arima_dt1)

  summary(inla_arima_dt1)

"inla(formula = y ~ lag_y + f(t, model = \"ar1\"), data = arima_dt1)"

Time used:
 Pre-processing    Running inla Post-processing           Total 
         0.3056          2.3598          0.1302          2.7956 

Fixed effects:
              mean     sd 0.025quant 0.5quant 0.975quant   mode kld
(Intercept) 0.0500 0.0328    -0.0104   0.0482     0.1196 0.0441   0
lag_y       0.4967 0.0896     0.3591   0.4733     0.6622 0.4615   0

Random effects:
Name	  Model
 t   AR1 model 

Model hyperparameters:
                                             mean        sd 0.025quant  0.5quant 0.975quant      mode
Precision for the Gaussian observations 1.872e+04 1.841e+04  1354.9766 1.333e+04  6.718e+04 3768.1493
Precision for t                         5.877e+00 1.238e+00     4.1083 5.644e+00  8.891e+00    5.1237
Rho for t                               6.334e-01 8.810e-02     0.4215 6.494e-01  7.612e-01    0.6879

Expected number of effective parameters(std dev): 998.53(1.969)
Number of equivalent replicates : 1.002 

Marginal log-Likelihood:  -303.71 

Neither the rho nor the lag_y coefficient are what I expected it to be... 

I guess my main question is: what's the closest implementation we can do to estimate an ARIMA model in INLA?

Thanks!

N

[INLA help]

The MA part is not implemented. Only AR processes

--
Håvard Rue
Helpdesk
he...@r-inla.org

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[Nafis Sadat]

Gotcha. Question though: If we were to add a lag term and an AR(1) model in an INLA call, what do the estimates mean in that case? In classic econometrics they keep referring to MA terms as lags on the residuals (autoregressive residuals if you will), and so I'm wondering if that fulfills the classic ARIMA model?

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[Helpdesk]

I would guess that they would be the same as the generalisation of the
AR(p), also using the marginal variance and not the innovation
variance, as its easier to set prior on the marginal properties

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