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Use D=1 and d=1. seasonal lags at 12,24,36. Now identify p,P and q,Q values from ACF and PACF graph. i am also confuse if ACF has spike at 11 should we consider it.

You have provided two graphs:

1. **ACF (Autocorrelation Function)** and
2. **PACF (Partial Autocorrelation Function)**

Based on the seasonal ARIMA model parameters you mentioned, where \( D = 1 \), \( d = 1 \), and seasonal lags occur at 12, 24, and 36, we need to extract the **AR (p), MA (q), Seasonal AR (P), and Seasonal MA (Q)** parameters by analyzing the ACF and PACF plots.

### Interpreting the ACF and PACF plots:

#### 1. **ACF (Autocorrelation Function) Plot:**
   - **ACF spikes:**
     - The ACF shows significant spikes at lags like 1, 2, 12, and possibly around 24. 
     - The presence of these spikes can indicate the moving average (MA) terms (for non-seasonal MA, \( q \)) and seasonal MA (\( Q \)).
     - The spike at lag 12 suggests a **seasonal MA term**, so \( Q \) could be 1.

#### 2. **PACF (Partial Autocorrelation Function) Plot:**
   - **PACF spikes:**
     - The PACF plot shows significant spikes at lags 1 and 12, which suggest **autoregressive (AR) terms**.
     - The spike at lag 12 indicates a **seasonal AR term**, so \( P \) could be 1.
     - For the non-seasonal AR (\( p \)), the spike at lag 1 suggests \( p = 1 \).

#### Addressing the spike at lag 11:
   - If there's a spike at lag 11, it may not directly contribute to the seasonality unless you have strong seasonal effects at intervals like 12 or multiples of 12. Hence, unless lag 11 stands out significantly (crossing confidence limits), we typically do not consider it for seasonal components.

### Conclusion:

Based on the interpretation of both plots:
- \( p = 1 \) (from PACF spike at lag 1)
- \( q = 1 \) (from ACF spike at lag 1)
- \( P = 1 \) (from PACF spike at lag 12 for seasonal AR)
- \( Q = 1 \) (from ACF spike at lag 12 for seasonal MA)

### Seasonal and Non-Seasonal Model: 
The model would be:
- **Non-Seasonal ARIMA**: \( p = 1 \), \( d = 1 \), \( q = 1 \)
- **Seasonal ARIMA**: \( P = 1 \), \( D = 1 \), \( Q = 1 \), with seasonal lag 12

Would you like further clarification on any of these points? 

Here are five additional questions to consider:

1. Would you like a deeper explanation of how seasonal lags impact ACF/PACF interpretations?
2. Should we try different values for the seasonal AR or MA terms based on different spikes?
3. Do you want to explore other methods for identifying AR and MA terms more precisely?
4. Are there additional steps you’d like to take to validate these parameter choices using diagnostic tests?
5. Would you like to understand how differencing impacts the ACF and PACF plots?

**Tip**: Seasonal ARIMA models often rely heavily on spikes at the seasonal lags (multiples of 12 for monthly data). Make sure these spikes are consistently strong to justify the seasonal terms.

Use D=1 and d=1. seasonal lags at 12,24,36. Now identify p,P and q,Q values from ACF and PACF graph. I am also confused if ACF has spike at 11, should we consider it?

Learn how to identify the AR (p), MA (q), seasonal AR (P), and seasonal MA (Q) parameters for a Seasonal ARIMA model using ACF and PACF graphs. The query involves understanding spikes at various lags, particularly at 1, 11, 12, 24, and 36, to build a robust seasonal ARIMA model. Suitable for advanced students and professionals in time series analysis.

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