Flexible HAR model for realized volatility

Francesco Audrino; Chen Huang; Ostap Okhrin

doi:10.1515/snde-2017-0080

Abstract

The Heterogeneous Autoregressive (HAR) model is commonly used in modeling the dynamics of realized volatility. In this paper, we propose a flexible HAR(1, . . . , p) specification, employing the adaptive LASSO and its statistical inference theory to see whether the lag structure (1, 5, 22) implied from an economic point of view can be recovered by statistical methods. The model differs from Audrino and Knaus (2016) [Audrino, F. and S. D. Knaus. 2016. “Lassoing the HAR model: A model selection perspective on realized volatility dynamics.” Econometrics Review 35: 1485–1521]. where the authors apply LASSO on the AR(p) model, which does not necessarily lead to a HAR model. Adaptive LASSO estimation and the subsequent hypothesis testing results fail to show strong evidence that such a fixed lag structure can be recovered by a flexible model. We also apply the group LASSO and related tests to check the validity of the classic HAR, which is rejected in most cases. The results justify our intention to use a flexible lag structure while still keeping the HAR frame. In terms of the out-of-sample forecasting, the proposed flexible specification works comparably to the benchmark HAR(1, 5, 22). Moreover, the time-varying model combinations show that when the market environment is not stable, the fixed lag structure (1, 5, 22) is not particularly accurate and effective.

Keywords: heterogeneous autoregressive model; realized volatility; lag structure; adaptive LASSO; hypothesis testing

JEL Classification: C12; C32; C51; C52

A Appendix

A.1 Supplementary Tables for Section 4.3

Table 5:

Descriptive statistics of the time series of RVt(d) for the 10 individual stocks and 19 indices data.

	mean⋅10²	std.⋅10²	min⋅10³	max	skewness	kurtosis	obs
Individual Stocks
BA	1.288	0.689	3.644	0.075	3.204	16.541	3013
IBM	1.023	0.615	3.145	0.102	4.549	35.388	3013
JNJ	0.818	0.443	2.398	0.068	4.440	33.995	3014
KO	0.902	0.474	2.311	0.071	4.164	29.759	3014
WMT	0.989	0.528	3.158	0.089	4.078	31.856	3013
CAT	1.474	0.881	3.984	0.127	3.414	19.876	3015
DIS	1.230	0.707	3.541	0.090	3.662	21.376	3013
NKE	1.119	0.680	3.181	0.133	5.294	57.956	3014
UNH	1.497	0.962	4.487	0.139	3.632	23.581	3015
XOM	1.119	0.680	3.181	0.133	5.292	54.917	3014
Indices
AEX	0.966	0.577	1.498	0.060	2.442	12.542	4387
All Ordinaries	0.604	0.347	1.113	0.039	2.685	15.456	4387
Bovespa	1.349	0.637	3.166	0.082	3.551	25.485	4387
CAC 40	1.060	0.588	2.017	0.072	2.588	16.122	4387
DAX	1.160	0.694	2.245	0.077	2.468	13.671	4387
DJIA	0.902	0.608	1.769	0.093	3.515	26.398	4387
Euro STOXX 50	1.132	0.672	0.216	0.104	3.068	22.602	4387
FT Straits Times	0.698	0.313	2.684	0.046	3.069	22.940	4387
FTSE 100	0.799	0.490	1.946	0.068	2.870	18.920	4387
FTSE MIB	1.017	0.569	2.593	0.073	2.374	14.283	4387
IBEX 35	1.106	0.566	2.025	0.074	2.170	14.456	4387
IPC Mexic	0.806	0.494	1.837	0.072	3.364	28.846	4387
KOSPI Composite	0.997	0.597	2.382	0.077	2.413	14.472	4387
Nasdaq 100	0.973	0.623	2.098	0.066	2.394	12.521	4386
Nikkei 225	0.947	0.483	2.293	0.057	2.816	18.231	4387
Russel 2000	0.906	0.550	1.713	0.076	3.057	19.573	4386
S&P 500	0.912	0.611	1.273	0.088	3.127	20.815	4387
S&P CNX Nifty	1.010	0.682	2.094	0.137	4.438	46.791	4387
Swiss Market	0.816	0.480	2.686	0.065	3.081	19.248	4387

Table 6:

In-sample RMSEs of each model (×103) for each stock/index.

	HAR (1, 5, 22)	HAR (a, b, c)	HAR-LASSO	AR-LASSO	AR-AIC	HAR-NP	HARQ	HARSJ	HARSJ-LASSO
Individual Stocks
BA	0.246	0.319	0.244	0.241	0.233	0.190	0.241	0.247	0.246
IBM	0.232	0.249	0.225	0.217	0.206	0.156	0.226	0.226	0.225
JNJ	0.145	0.163	0.147	0.145	0.140	0.098	0.140	0.147	0.149
KO	0.133	0.130	0.123	0.125	0.119	0.079	0.130	0.133	0.129
WMT	0.178	0.176	0.165	0.167	0.161	0.094	0.174	0.177	0.173
CAT	0.382	0.387	0.370	0.369	0.353	0.276	0.369	0.378	0.375
DIS	0.278	0.339	0.273	0.271	0.263	0.213	0.270	0.278	0.285
NKE	0.192	0.231	0.184	0.185	0.208	0.115	0.187	0.181	0.183
UNH	0.417	0.497	0.406	0.408	0.394	0.305	0.416	0.425	0.421
XOM	0.356	0.361	0.344	0.339	0.327	0.151	0.344	0.359	0.357
Indices
AEX	0.122	0.131	0.113	0.114	0.115	0.089	–	0.118	0.113
All Ordinaries	0.049	0.055	0.049	0.050	0.048	0.042	–	0.048	0.049
Bovespa	0.225	0.224	0.204	0.202	0.212	0.164	–	0.216	0.196
CAC 40	0.142	0.152	0.140	0.142	0.134	0.093	–	0.137	0.140
DAX	0.178	0.188	0.169	0.169	0.163	0.119	–	0.172	0.165
DJIA	0.149	0.160	0.159	0.158	0.138	0.095	–	0.143	0.154
Euro STOXX 50	0.206	0.211	0.206	0.204	0.194	0.113	–	0.195	0.205
FT Straits Times	0.043	0.052	0.040	0.041	0.040	0.030	–	0.042	0.042
FTSE 100	0.088	0.091	0.087	0.086	0.083	0.057	–	0.086	0.087
FTSE MIB	0.117	0.130	0.117	0.116	0.114	0.083	–	0.112	0.114
IBEX 35	0.137	0.143	0.135	0.136	0.132	0.104	–	0.130	0.131
IPC Mexico	0.074	0.075	0.073	0.073	0.071	0.063	–	0.071	0.073
KOSPI Composite	0.121	0.118	0.114	0.115	0.111	0.072	–	0.113	0.115
Nasdaq 100	0.108	0.113	0.098	0.100	0.100	0.077	–	0.101	0.099
Nikkei 225	0.114	0.116	0.101	0.102	0.105	0.074	–	0.112	0.101
Russel 2000	0.127	0.134	0.124	0.124	0.118	0.096	–	0.119	0.119
S&P 500	0.149	0.162	0.143	0.144	0.139	0.095	–	0.139	0.141
S&P CNX Nifty	0.219	0.235	0.223	0.219	0.215	0.129	–	0.208	0.217
Swiss Market	0.085	0.091	0.084	0.084	0.080	0.063	–	0.081	0.082

The ratio in bold implies the model performs the best in in-sample fitting among all models.

Table 7:

Out-of-sample RMSEs of each model (×103) for each stock/index.

	HAR (1, 5, 22)	HAR (a, b, c)	HAR-LASSO	AR-LASSO	AR-AIC	HAR-NP	HARQ	HARSJ	HARSJ-LASSO
Individual Stocks
BA	0.284	0.326	0.303	0.297	0.305	0.378	0.315	0.409	0.294
IBM	0.309	0.311	0.316	0.387	0.325	0.264	0.305	0.364	0.309
JNJ	0.187	0.189	0.184	0.192	0.189	0.181	0.211	0.231	0.190
KO	0.176	0.167	0.178	0.183	0.183	0.280	0.172	0.235	0.175
WMT	0.265	0.265	0.257	0.270	0.235	0.221	0.334	0.319	0.248
CAT	0.531	0.510	0.526	0.567	0.529	0.691	0.533	0.676	0.480
DIS	0.343	0.397	0.366	0.354	0.355	0.456	0.495	0.517	0.407
NKE	0.681	0.691	0.710	0.698	0.692	1.858	0.754	1.221	0.698
UNH	0.697	0.746	0.728	0.710	0.691	0.748	3.061	1.650	1.285
XOM	0.535	0.544	0.546	0.537	0.488	0.294	1.508	0.588	0.565
Indices
AEX	0.129	0.135	0.128	0.129	0.129	0.176	–	0.222	0.126
All Ordinaries	0.063	0.068	0.066	0.068	0.065	0.086	–	0.083	0.064
Bovespa	0.277	0.277	0.266	0.270	0.305	0.341	–	0.376	0.249
CAC 40	0.164	0.173	0.181	0.182	0.160	0.224	–	0.237	0.178
DAX	0.194	0.198	0.205	0.193	0.200	0.227	–	0.284	0.191
DJIA	0.235	0.233	0.247	0.225	0.227	0.405	–	0.269	0.235
Euro STOXX 50	0.296	0.294	0.311	0.344	0.270	0.253	–	0.341	0.315
FT Straits Times	0.060	0.071	0.049	0.051	0.061	0.068	–	0.094	0.051
FTSE 100	0.099	0.102	0.102	0.105	0.095	0.138	–	0.101	0.097
FTSE MIB	0.138	0.147	0.138	0.147	0.141	0.170	–	0.240	0.151
IBEX 35	0.184	0.188	0.182	0.183	0.175	0.238	–	0.246	0.181
IPC Mexico	0.095	0.093	0.093	0.096	0.096	0.120	–	0.118	0.097
KOSPI Composite	0.164	0.151	0.138	0.148	0.158	0.168	–	0.218	0.140
Nasdaq 100	0.141	0.151	0.132	0.126	0.142	0.242	–	0.181	0.118
Nikkei 225	0.146	0.147	0.136	0.137	0.156	0.173	–	0.186	0.134
Russel 2000	0.181	0.186	0.174	0.169	0.177	0.313	–	0.209	0.150
S&P 500	0.239	0.229	0.217	0.203	0.224	0.169	–	0.242	0.192
S&P CNX Nifty	0.338	0.302	0.296	0.399	0.259	0.266	–	0.369	0.275
Swiss Market	0.100	0.104	0.100	0.101	0.101	0.142	–	0.146	0.106

The ratio in bold implies the model performs the best in out-of-sample forecasting among all models.

A.2 Robustness check

Here we deliver the robustness check results of other approach in determining the tuning parameter. Alternatively, choosing λ based on the Bayesian Information Criterion (BIC) is also commonly used in empirical studies due to its computational simplicity. We compare the in-sample fitting and out-of-sample forecasting performance of the HAR-LASSO model with the hν-block cross-validation and BIC. The results are presented by the RMSE ratios relative to classic HAR(1, 5, 22) in Table 8.

Table 8:

In- and out-of-sample RMSE ratios relative to HAR(1, 5, 22), for HAR-LASSO model choosing tuning parameter with hν-block cross-validation and BIC.

		HAR(1, 5, 22)	HAR-LASSO (hν-block)	HAR-LASSO (BIC)
In-sample fitting
Stocks	Mean	1.0000	0.9546	0.9294
	Median	1.0000	0.9571	0.9375
Indices	Mean	1.0000	0.9668	0.9521
	Median	1.0000	0.9680	0.9484
Out-of-sample forecasting
Stocks	Mean	1.0000	1.0022	1.3802
	Median	1.0000	1.0101	1.2673
Indices	Mean	1.0000	1.0119	1.1498
	Median	1.0000	1.0136	1.1192

Table 8 shows that determining λ with BIC makes the in-sample fitting of the flexible HAR model more accurate. However, we found that BIC yields quite bad forecasting performance in terms of out-of-sample.

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Supplementary Material

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Published Online: 2018-11-15

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