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I am trying to hand calculate the prediction output for statsmodel's SARIMAX but am not getting the right values. I fit the model as follows:

from statsmodels.tsa.statespace.sarimax import SARIMAX
import pandas as pd
import numpy as np

x = np.array([1.43839683, 1.58737972, 2.56918062, 2.20768073, 2.06686168,
       1.79483696, 2.10348052, 2.24404145, 1.38084798, 1.8165772 ,
       2.23706359, 2.06938327, 1.45480011, 1.59935103, 2.56467497,
       2.19698115, 2.05029322, 1.77362942, 2.08021718, 2.21928505,
       1.3667532 , 1.80440352, 2.23990379, 2.08945659, 1.49048437,
       1.65989372, 2.63922424, 2.29742929, 2.17817642, 1.9274815 ,
       2.25526355, 2.40683635, 1.57503051, 2.01350068, 2.45757963,
       2.31875038, 1.72791459, 1.91297156, 2.88799343, 2.5434238 ,
       2.41232631, 2.14680717, 2.45679305, 2.58724137, 1.7465048 ,
       2.15493785, 2.56915222, 2.39105932, 1.75722185, 1.92676286,
       2.89753149, 2.56273977, 2.43401002, 2.17312866, 2.48684727,
       2.62114427, 1.79892782, 2.21660381, 2.65270203, 2.50424027])

model = SARIMAX(
    x,
    order=(1, 0, 3),
    seasonal_order=(1, 0, 1, 12),
).fit()

df = pd.DataFrame({
    "x": x,
    "pred": model.predict(),
    "resid": model.resid
})
df.head()

|    |       x |    pred |     resid |
|---:|--------:|--------:|----------:|
|  0 | 1.4384  | 0       |  1.4384   |
|  1 | 1.58738 | 1.43838 |  0.149002 |
|  2 | 2.56918 | 1.61701 |  0.952175 |
|  3 | 2.20768 | 2.76155 | -0.553871 |
|  4 | 2.06686 | 2.20161 | -0.134748 |

To calculate the forecasts by hand, I first calculate the terms separately as follows:

df["hand_calc"] = 0

# Add AR terms
for i, v in enumerate(model.arparams):
    df[f"ar_{i+1}_term"] = v * df.x.shift(i+1)
    df["hand_calc"] += df[f"ar_{i+1}_term"]

# Add Seasonal AR terms
for i, v in enumerate(model.seasonalarparams):
    df[f"ar_S{i+1}_term"] = v * df.x.shift(12 * (i+1))
    df["hand_calc"] += df[f"ar_S{i+1}_term"]

# Add MA terms
for i, v in enumerate(model.maparams):
    df[f"ma_{i+1}_term"] = v * df.resid.shift(i+1)
    df["hand_calc"] += df[f"ma_{i+1}_term"]
    
# Add Seasonal MA terms
for i, v in enumerate(model.seasonalmaparams):
    df[f"ma_S{i+1}_term"] = v * df.resid.shift(12*(i+1))
    df["hand_calc"] += df[f"ma_S{i+1}_term"]
    
df["pred_diff"] = df.pred - df.hand_calc

df.tail()

|    |       x |    pred |      resid |   hand_calc |   ar_1_term |   ar_S1_term |   ma_1_term |    ma_2_term |   ma_3_term |   ma_S1_term |   pred_diff |
|---:|--------:|--------:|-----------:|------------:|------------:|-------------:|------------:|-------------:|------------:|-------------:|------------:|
| 55 | 2.62114 | 2.59697 | 0.0241698  |     5.05791 |     2.47886 |      2.58496 |  0.00450707 | -0.000666605 | -0.00223555 |  -0.00751738 |    -2.46093 |
| 56 | 1.79893 | 1.77915 | 0.0197747  |     4.36447 |     2.61272 |      1.74497 |  0.0091306  |  0.00507481  | -0.0015117  |  -0.00591724 |    -2.58531 |
| 57 | 2.2166  | 2.21186 | 0.00473958 |     3.95877 |     1.79315 |      2.15304 |  0.00747028 |  0.0102807   |  0.0115084  |  -0.0166786  |    -1.74691 |
| 58 | 2.6527  | 2.63771 | 0.01499    |     4.79474 |     2.20948 |      2.56689 |  0.00179047 |  0.00841128  |  0.0233142  |  -0.0151472  |    -2.15703 |
| 59 | 2.50424 | 2.47054 | 0.0336958  |     5.03754 |     2.64418 |      2.38895 |  0.00566277 |  0.00201601  |  0.0190747  |  -0.0223449  |    -2.567 

I have tried the same calculations without the seasonal terms but still not able to replicate the predictions.

In addition, I am not sure how to make out of sample predictions given that we do not have MA terms available.

I looked through the following solutions but was still not successful:

ARIMA SARIMA model mathematical formula

and

Unable to recreate Statsmodels ARIMAX (1, 1, 0) forecasts by hand

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  • $\begingroup$ Some points I found: 1. only AR models (SARIMAX(1,0,0)(0,0,0,12)) give me perfect replication. 2. If i take SARIMAX(1,0,1)(0,0,0,12), I get errors but they reduce over time $\endgroup$ – hamiq Nov 16 '20 at 23:30
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    $\begingroup$ This not how a SARIMA works when you have both components. You need to multiply the two polynomials for the observation AR and then seasonal AR to get a pure observation time AR. Same for the MA when you have both components. The reason why your previous calculations worked is that you only have 1 component at a time. For example, your AR part in pure observation time is (1-.9.967L-.997940L^{12} + 0.99609189L^(13))Y(t) $\endgroup$ – Kevin S Nov 17 '20 at 0:10
  • $\begingroup$ Hi Kevin, thanks for pointing out the mistake. I tried with a simple SARIMAX(1,0,1)(0,0,0) and still am not getting the calculations right. I added the seasonal terms hoping that someone would show the full set of calculations $\endgroup$ – hamiq Nov 17 '20 at 9:09
  • $\begingroup$ Basically the issue is that the moving-average term in SARIMAX models must be estimated, and so it is not identical to the resid attribute near the beginning of the sample. See my answer to stats.stackexchange.com/questions/430186/…. I don't know of a simpler method for computing the exact results than actually applying the Kalman filter recursions. $\endgroup$ – cfulton Nov 17 '20 at 12:20
  • $\begingroup$ @cfulton what would be the steps required to compute the exact solution? Also, are MA terms estimated for k-step ahead forecasts (into the future) or do we assume them to be zero? $\endgroup$ – hamiq Nov 17 '20 at 13:41

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