Nonparametric GARCH: An Iterated Neural Estimator

Ruizhi Deng; Guohao Shen; Ngai Hang Chan

stat.MLNov 3, 2026

Nonparametric GARCH: An Iterated Neural Estimator

Authors: Ruizhi Deng, Guohao Shen, Ngai Hang Chan

Organizations: Department of Applied Mathematics, The Hong Kong Polytechnic University · Department of Biostatistics, City University of Hong Kong

Abstract

We study nonparametric estimation and forecasting for GARCH-type processes and propose an iterated neural estimator that alternates between estimating the recursion map and updating the unobserved conditional variances. Under mild conditions, we establish finite-sample bounds for the time-averaged in-sample variance-path error and show that the estimator is minimax rate-optimal up to logarithmic factors. Under stationarity and mixing conditions, we establish the same rate for recursion-map estimation and derive finite-sample bounds for out-of-sample variance forecasts and expected excess Gaussian quasi-likelihood (QLIKE) loss. The convergence rates depend on the true lag orders and remain unchanged when the fitted orders are overspecified. Simulation studies illustrate the effectiveness of our estimators under nonlinear and time-varying volatility dynamics. Real-data applications suggest that our neural methods perform competitively and can incorporate option-price and high-frequency information.

Figures & tables

Panel of Figure 1 — Figure 1: The neural network used at each iteration. The inputs are the $p$ return lags and the $q$ variance estimates from the preceding iteration. The $\mathcal{D}$ hidden layers have maximum width $\mathcal{W}$ and map the inputs to a positive conditional-variance estimate.

Panel A.	Parametric designs			Nonlinear designs
	A(2,2)	B(2,2)	C(2,2)	D(3,3)	E(3,4)	F(4,3)	G(4,4)	H(5,5)
MLE	0.08 [-1pt] (0.12)	0.93 [-1pt] (0.21)	1.77 [-1pt] (0.22)	8.04 [-1pt] (0.71)	8.93 [-1pt] (2.01)	12.09 [-1pt] (0.99)	10.36 [-1pt] (1.09)	3.77 [-1pt] (0.25)
EGARCH	0.72 [-1pt] (1.55)	0.49 [-1pt] (0.32)	0.08 [-1pt] (0.08)	3.28 [-1pt] (0.53)	4.86 [-1pt] (1.58)	10.84 [-1pt] (0.81)	4.65 [-1pt] (1.42)	3.47 [-1pt] (0.21)
GJR	0.15 [-1pt] (0.16)	0.09 [-1pt] (0.06)	0.51 [-1pt] (0.46)	6.89 [-1pt] (1.25)	9.00 [-1pt] (2.26)	12.53 [-1pt] (1.00)	14.51 [-1pt] (5.33)	3.88 [-1pt] (0.26)
Kernel	4.35 [-1pt] (5.12)	2.65 [-1pt] (1.53)	1.23 [-1pt] (0.44)	3.80 [-1pt] (0.96)	3.59 [-1pt] (0.97)	7.59 [-1pt] (1.22)	4.92 [-1pt] (0.91)	2.61 [-1pt] (0.47)
NN-MLP	1.62 [-1pt] (2.15)	0.86 [-1pt] (1.27)	0.57 [-1pt] (0.22)	2.92 [-1pt] (0.57)	2.14 [-1pt] (0.40)	5.13 [-1pt] (0.76)	2.31 [-1pt] (0.54)	2.21 [-1pt] (0.29)

Table 1: Out-of-sample performance under stationary dynamics. Panel A reports variance-path MSE, and Panel B reports latent excess QLIKE. Entries are medians over 50 replications, with Monte Carlo standard deviations in parentheses. Values are scaled by 100.

Figure 2: Lag-order sensitivity under design G $(4,4)$ . Cell entries are median out-of-sample variance-path MSEs in units of $10^{-2}$ over 20 replications. Dashed lines identify fitted orders below the structural values, and boxes mark the structural order. The shaded area in the guide marks fitted orders below at least one structural value.

Metric	Design	MLE	EGARCH	GJR	Kernel	NN- MLP	NN- Linear	NN- EW	NN- Window	LSTM
Future path MSE	Scale	18.01 [-1pt] (5.17)	11.32 [-1pt] (2.18)	18.11 [-1pt] (5.80)	16.83 [-1pt] (2.70)	6.50 [-1pt] (1.69)	5.54 [-1pt] (1.97)	5.38 [-1pt] (2.18)	4.79 [-1pt] (2.11)	5.03 [-1pt] (2.09)
	Function	2.58 [-1pt] (0.68)	1.79 [-1pt] (0.40)	3.01 [-1pt] (0.77)	2.13 [-1pt] (0.34)	1.43 [-1pt] (0.35)	1.02 [-1pt] (0.36)	0.97 [-1pt] (0.40)	1.28 [-1pt] (0.53)	1.27 [-1pt] (0.34)
	Joint	16.69 [-1pt] (5.06)	21.05 [-1pt] (6.01)	17.51 [-1pt] (5.12)	23.85 [-1pt] (6.41)	9.89 [-1pt] (2.69)	9.16 [-1pt] (2.51)	8.88 [-1pt] (2.34)	7.04 [-1pt] (2.51)	6.77 [-1pt] (2.65)
Latent excess QLIKE	Scale	8.89 [-1pt] (1.35)	4.59 [-1pt] (0.84)	8.69 [-1pt] (1.41)	35.98 [-1pt] (7.47)	11.27 [-1pt] (3.13)	7.56 [-1pt] (2.53)	5.60 [-1pt] (2.05)	3.11 [-1pt] (1.38)	5.44 [-1pt] (3.61)
	Function	4.90 [-1pt] (0.67)	4.42 [-1pt] (0.77)	5.36 [-1pt] (0.79)	8.40 [-1pt] (1.79)	4.99 [-1pt] (1.12)	2.83 [-1pt] (1.51)	2.29 [-1pt] (0.82)	2.52 [-1pt] (4.68)	3.34 [-1pt] (0.99)
	Joint	5.16 [-1pt] (1.20)	4.55 [-1pt] (0.52)	5.48 [-1pt] (1.23)	24.24 [-1pt] (7.72)	6.61 [-1pt] (2.38)	5.65 [-1pt] (1.89)	4.74 [-1pt] (1.46)	2.38 [-1pt] (1.39)	4.39 [-1pt] (3.24)

Table 2: Out-of-sample performance under temporal change. The two blocks report variance-path MSE and latent excess QLIKE. Entries are medians over 50 replications, with Monte Carlo standard deviations in parentheses. Values are scaled by 100.

Figure 3: Running average test QLIKE for the eight-stock daily panel. Each curve first averages QLIKE equally across stocks on each date and then averages through the corresponding forecast date.

Method	MMM	UNP	IBM	NKE	NEE	TGT	SBUX	KHC
GARCH	2.26 [-1pt] (57.26)	1.64 [-1pt] (12.91)	2.28 [-1pt] (78.95)	2.95 [-1pt] (597.44)	2.03 [-1pt] (12.94)	2.80 [-1pt] (567.18)	2.71 [-1pt] (358.59)	1.71 [-1pt] (11.38)
EGARCH	2.22 [-1pt] (54.06)	1.62 [-1pt] (12.77)	2.24 [-1pt] (77.26)	2.94 [-1pt] (1018.43)	2.03 [-1pt] (12.76)	2.82 [-1pt] (5549.27)	2.65 [-1pt] (341.79)	1.72 [-1pt] (11.49)
GJR	2.24 [-1pt] (55.34)	1.59 [-1pt] (12.24)	2.28 [-1pt] (79.78)	2.93 [-1pt] (622.61)	2.06 [-1pt] (13.88)	2.85 [-1pt] (715.46)	2.80 [-1pt] (361.26)	1.73 [-1pt] (12.04)
Kernel	2.39 [-1pt] (54.14)	1.72 [-1pt] (14.48)	2.34 [-1pt] (78.62)	3.05 [-1pt] (571.12)	2.05 [-1pt] (13.10)	2.86 [-1pt] (514.95)	2.77 [-1pt] (344.04)	1.70 [-1pt] (11.23)
NN-MLP	2.27 [-1pt] (53.80)	1.74 [-1pt] (14.44)	2.31 [-1pt] (78.80)	2.78 [-1pt] (564.27)	2.02 [-1pt] (12.85)	2.64 [-1pt] (509.84)	2.55 [-1pt] (341.19)	1.69 [-1pt] (10.90)
NN-EW	2.30 [-1pt] (53.28)	1.73 [-1pt] (14.38)	2.48 [-1pt] (79.67)	2.72 [-1pt] (562.52)	2.02 [-1pt] (12.53)	2.69 [-1pt] (509.62)	2.57 [-1pt] (341.55)	1.71 [-1pt] (11.02)

Table 3: Test QLIKE and variance MSE for eight stocks. Each entry reports QLIKE, with variance MSE in parentheses. Both values are rounded to two decimal places.

Training response	Method	M0	M1	M2
$Y_{t}^{2}$	Kernel	2.724 (397.67)	2.556 (395.91)	2.496 (386.86)
	NN-MLP	2.761 (394.98)	2.467 (383.69)	2.472 (384.36)
	NN-EW	2.812 (396.94)	2.566 (389.93)	2.575 (389.38)
	LSTM	2.697 (394.95)	2.682 (394.35)	2.682 (394.16)
$RV_{t}$	Kernel	2.611 (393.21)	2.478 (389.54)	2.490 (390.45)
	NN-MLP	2.602 (392.24)	2.423 (378.37)	2.416 (372.00)

Table 4: Average next-day realized-variance forecast errors for the eight stocks under three information sets. Entries report mean QLIKE across stocks, with mean variance MSE in parentheses. All specifications use the same 155 test dates.

	MMM		UNP
Method	Baseline	Augmented	Baseline	Augmented
Kernel	6.0178 (0.8866)	6.0427 (0.8948)	5.6890 (0.5308)	5.6560 (0.5157)
NN-MLP	6.0142 ( 0.8526 )	5.9850 ( 0.8245 )	5.6249 ( 0.4992 )	5.6081 (0.4928)
NN-EW	6.0661 (0.8711)	5.9682 (0.8307)	5.7167 (0.5273)	5.6059 ( 0.4919 )
LSTM	6.0100 (0.8670)	6.0189 (0.8430)	5.6801 (0.5169)	5.6415 (0.4966)

Table 5: Test losses for five-minute forecasts of MMM and UNP. Entries report QLIKE, with variance MSE in millions in parentheses. Both stocks have 200 common test days and 72 evaluated intervals per day.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

DGP	$T$	MLE	EGARCH	GJR	Kernel	NN-MLP	LSTM
A $(2,2)$	2000	0.38 (0.57)	1.30 (2.14)	0.62 (0.63)	8.05 (9.03)	4.45 (12.85)	2.22 (7.89)
	4000	0.20 (0.20)	1.22 (1.26)	0.31 (0.24)	7.11 (4.39)	2.50 (5.47)	1.81 (3.53)
	8000	0.09 (0.10)	1.11 (0.76)	0.16 (0.12)	5.97 (2.28)	1.95 (2.12)	1.54 (1.95)
B $(2,2)$	2000	1.07 (0.27)	0.63 (0.35)	0.30 (0.20)	4.00 (1.78)	2.11 (2.65)	0.78 (1.94)
	4000	0.99 (0.17)	0.55 (0.23)	0.17 (0.13)	3.43 (1.04)	1.37 (0.99)	0.76 (1.43)
	8000	0.97 (0.15)	0.52 (0.22)	0.10 (0.06)	2.83 (0.87)	1.00 (1.52)	0.74 (0.67)

Appendix

Table 6: In-sample variance-path MSE by sample size. Medians (Monte Carlo standard deviations) over 50 replications, scaled by 100.

	MMM	UNP	IBM	NKE	NEE	TGT	SBUX	KHC
BDS	4.1144	3.4150	3.2088	2.9916	2.8526	2.8237	2.7894	2.7473
Auxiliary	IBM	MMM	MMM	SBUX	SBUX	NKE	NKE	MMM
Correlation	0.5694	0.5516	0.5694	0.5135	0.4128	0.3494	0.5135	0.3939

Appendix

Table 8: Stock selection for daily data. Auxiliary stocks are selected by daily-return correlations over 2016–2021.

Figure S.1: NN-MLP lag-order sensitivity under nonlinear stationary dynamics. Each cell reports the median out-of-sample variance-path MSE over 20 replications, scaled by 100. The orange box marks the structural lag orders, and the dashed lines mark the corresponding return- and variance-lag boundaries.

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