I'm working with the Australian Red Wine Data:

QUESTION 1: What model should I try next? 

After I did a Log Transformation and Differenced at lag 12 and lag 1, here is the sample ACF/PACF I got:

Here's what I've done so far:

1. I ran an autofit;

2. I set some insignificant parameters to zero;

3. I ran MLE estimation. 

Here is the result (MA(13) with 6 non-zero parameters):

Method: Maximum Likelihood

ARMA Model:

X(t) = Z(t) - .7764 Z(t-1) + .0000 Z(t-2) + .0000 Z(t-3)

     + .0000 Z(t-4) + .0000 Z(t-5) - .1306 Z(t-6) + .0000 Z(t-7)

     + .1785 Z(t-8) + .0000 Z(t-9) + .0000 Z(t-10) - .1217 Z(t-11)

     - .6370 Z(t-12) + .4805 Z(t-13)

WN Variance = .011647

MA Coefficients

      -.776414       .000000       .000000       .000000

       .000000      -.130596       .000000       .178469

       .000000       .000000      -.121740      -.637029

       .480473

Standard Error of MA Coefficients

       .071543       .000000       .000000       .000000

       .000000       .088887       .000000       .081879

       .000000       .000000       .098314       .132459

       .102127

(Residual SS)/N = .0116471

AICC = -.178970E+03  

BIC  = -.182171E+03  

-2Log(Likelihood) = -.193895E+03

Accuracy parameter = .00199000

Number of iterations = 10

Number of function evaluations = 107

Optimization stopped within accuracy level.

QUESTION 2:  Power Transformation

In the very beginning I ran a log transformation, I also tried box-cox to get the optimal power transformation and here is the result: 

bcPower Transformation to Normality

      Est.Power Std.Err. Wald Lower Bound Wald Upper Bound

Y1 0.413         0.2085           0.0043                   0.8218

Likelihood ratio tests about transformation parameters

LRT df pval

LR test, lambda = (0) 4.045694 1 0.044284174

LR test, lambda = (1) 7.576282 1 0.005914129

It looks like I should choose lambda to be 0.413 (or the nearest interpretable 0.5); But when I fit an auto.arima to the transformed data, I got a worse AIC than the outcome of log transformation. 

QUESTION 3: Inconsistency in Heteroscedasticity Tests:

I ran a test for heteroscedasticity on my model residuals; and I ran an ARCHtest in the very beginning(before log transformation):

Non-constant Variance Score Test

Variance formula: ~ fitted.values

Chisquare = 0.01648741 Df = 1 p = 0.8978298

ARCH LM-test; Null hypothesis: no ARCH effects

data: wine

Chi-squared = 104.33, df = 12, p-value < 2.2e-16

Should I try ARCH Garch model or should I believe in homoscedasticity and keep the log transformation?