Any-mean-model + GARCH(1, 1) for probabilistic stock forecasting

In this blog post, we will explore the combination of any model with GARCH(1,1) for probabilistic stock forecasting. This approach allows us to capture both the conditional mean and conditional variance of stock returns. We will demonstrate the implementation using Python and the ahead package.

Ref: https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity

See also:

• https://thierrymoudiki.github.io/blog/2025/06/21/r/python/beyond-garch-statistical
• https://thierrymoudiki.github.io/blog/2025/06/03/python/beyond-garch-python
• https://thierrymoudiki.github.io/blog/2025/06/02/r/beyond-garch

install.packages("pak")
pak::pak("fpp2")
devtools::install_github("Techtonique/ahead")

pak::pak("fGarch")

(res <- ahead::agnosticgarchf(fpp2::goog200,
                           FUN=forecast::auto.arima, h=20))

    Point Forecast    Lo 95    Hi 95
201       532.1750 518.1879 546.1621
202       532.8717 518.8757 546.8678
203       533.5684 519.5635 547.5734
204       534.2652 520.2513 548.2790
205       534.9619 520.9391 548.9847
206       535.6586 521.6269 549.6903
207       536.3553 522.3148 550.3959
208       537.0521 523.0026 551.1015
209       537.7488 523.6905 551.8071
210       538.4455 524.3783 552.5127
211       539.1422 525.0662 553.2183
212       539.8390 525.7540 553.9239
213       540.5357 526.4419 554.6295
214       541.2324 527.1298 555.3351
215       541.9291 527.8176 556.0407
216       542.6259 528.5055 556.7462
217       543.3226 529.1934 557.4518
218       544.0193 529.8813 558.1573
219       544.7160 530.5692 558.8629
220       545.4128 531.2571 559.5684

ggplot2::autoplot(res)

(res <- ahead::agnosticgarchf(fpp2::goog200,
                           FUN=forecast::thetaf, h=20))

    Point Forecast    Lo 95    Hi 95
201       531.4982 518.2318 544.7646
202       531.7610 516.1991 547.3230
203       532.0238 514.1041 549.9436
204       532.2867 511.9078 552.6655
205       532.5495 509.5788 555.5201
206       532.8123 507.0890 558.5356
207       533.0751 504.4117 561.7384
208       533.3379 501.5205 565.1553
209       533.6007 498.3883 568.8131
210       533.8635 494.9868 572.7402
211       534.1263 491.2863 576.9663
212       534.3891 487.2548 581.5234
213       534.6519 482.8584 586.4454
214       534.9147 478.0601 591.7693
215       535.1775 472.8202 597.5348
216       535.4403 467.0955 603.7851
217       535.7031 460.8392 610.5670
218       535.9659 454.0004 617.9315
219       536.2288 446.5235 625.9340
220       536.4916 438.3483 634.6349

ggplot2::autoplot(res)

(res <- ahead::agnosticgarchf(fpp2::goog200,
                           FUN=ahead::ridge2f, h=20))

    Point Forecast    Lo 95    Hi 95
201       532.1740 519.4742 544.8737
202       532.9003 517.9900 547.8106
203       533.6547 516.8217 550.4877
204       534.4346 515.8770 552.9921
205       535.2377 515.1028 555.3726
206       536.0622 514.4648 557.6596
207       536.9062 513.9392 559.8731
208       537.7679 513.5087 562.0272
209       538.6460 513.1598 564.1321
210       539.5390 512.8824 566.1956
211       540.4458 512.6679 568.2236
212       541.3651 512.5096 570.2207
213       542.2961 512.4018 572.1905
214       543.2378 512.3395 574.1361
215       544.1894 512.3188 576.0601
216       545.1501 512.3359 577.9643
217       546.1193 512.3879 579.8506
218       547.0963 512.4721 581.7205
219       548.0805 512.5859 583.5751
220       549.0716 512.7274 585.4158

ggplot2::autoplot(res)

(res <- ahead::agnosticgarchf(fpp2::goog200,
                           FUN=ahead::loessf, h=20))

    Point Forecast    Lo 95    Hi 95
201       544.3276 520.1558 568.4994
202       548.2888 523.7199 572.8578
203       552.0027 527.0698 576.9356
204       554.1810 528.9140 579.4481
205       556.8539 531.2797 582.4282
206       560.3601 534.5032 586.2171
207       561.5067 535.3893 587.6240
208       565.6928 539.3353 592.0502
209       567.2796 540.7006 593.8586
210       570.1362 543.3525 596.9198
211       574.0123 547.0395 600.9850
212       578.5510 551.4034 605.6986
213       580.8285 553.5191 608.1379
214       584.2104 556.7512 611.6695
215       587.9165 560.3187 615.5144
216       591.1329 563.4066 618.8593
217       594.0383 566.1929 621.8837
218       595.6430 567.6872 623.5988
219       602.2172 574.1590 630.2754
220       604.7474 576.5943 632.9006

ggplot2::autoplot(res)

As we can see from the plots, combining any model with GARCH(1,1) provides a comprehensive view of both the expected stock prices and the associated uncertainty. This method is particularly useful for financial forecasting, where volatility plays a significant role.