A reality check on the GARCH-MIDAS volatility models

Abstract

We employ a battery of model evaluation tests for a broad set of GARCH-MIDAS models and account for data snooping bias. We document that inferences based on standard tests for GM variance components can be misleading. Our data mining free results show that the gain of macro-variables in forecasting total (long-run) variance by GM models is overstated (understated). Estimation of different components of volatility is crucial for designing differentiated investing strategies, risk management plans and pricing derivative securities. Therefore, researchers and practitioners should be wary of data-mining bias, which may contaminate a forecast that may appear statistically validated using robust evaluation tests.

KEYWORDS:

• Forecasting
• GARCH-MIDAS models
• component variance forecasts
• macro-variables
• data snooping

JEL Classifications:

• C32
• C52
• G11
• G17

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 The multiplicative component structure of these models combines the GARCH volatility with the long-run approximations for long-run variance – estimated by the innovative Mixed Data Sampling (MIDAS) approach of Ghysels, Santa-Clara, and Valkanov (2004, 2006). Given the importance of long-memory dependence in financial market volatility, the long-run variance smoothing by the MIDAS regressions – forecasting mismatched time frequency (e.g. monthly) variance from high frequency data points such as past daily squared returns or economic variables – has opened up new frontiers in examining the role of long memory processes in shaping financial volatility and how cross-market long-run variances and correlations depict inter- and intra-market integration patterns.

2 We invariably refer long-term variance to trend/secular/MIDAS component or variance to imply the same throughout the paper.

3 In addition to modelling conditional variance, the extensions that build on the GM framework have been utilised in studying the relationships between oil and stock market volatilities, oil-stock correlation, stock-bond correlation, oil-macroeconomic relationships, European equity market integration patterns and the effect of investor sentiment on US stock-bond correlation patterns (Conrad, Loch, and Rittler 2014; Asgharian, Christiansen, and Hou 2016; Pan et al. 2017; Virk and Javed 2017;Fang, Yu, and Huang 2018).

4 There are numerous stylized facts documented about financial market volatility, such as clustering, leverage effect, mean reversion and co-movement of volatilities among assets and across markets (Cont 2001).

5 Potentially, daily RV can be a noisy estimate for forecasting volatility at high frequencies (Andersen and Bollerslev 1997, 1998). However, we argue that the same is not applicable to the low frequency RV estimates when we model low frequency variance components at monthly, quarterly and bi-annual along with the total variance forecasts.

6 To this extent, Andersen and Bollerslev (1998) show that a well-specified, GARCH-type volatility filter smooth unconditional realised variance/volatility (RV), from high frequency intraday data, produces precise inter-daily predictions i.e., a latent volatility factor from high frequency intraday data improves out-of-sample forecasts at inter-daily variance predictions.

7 Although the importance of economic sources influencing market volatility is undeniable, there are limitations in the modelling the contribution of economic variables to the total and long-run variance forecasts. These include identification of the aggregate variable(s) that contributes to the evolution of financial market variance and prediction of financial market volatility on the basis of an information set that is prone to measurement errors and revisions compared with other variables to proxy long-run variations in the conditional variance such as RV.

8 This applies to tests for both total variance and long-run variance components regardless of whether the benchmark model is the GM model that smooths realised variance in MIDAS regressions (Engle, Ghysels, and Sohn 2013 and Conrad and Loch 2015) or the baseline GARCH (1, 1) specification (Asgharian, Hou, and Javed 2013 and Conrad and Loch 2015).

9 The pairwise DM test examines the equal predictability (EPA) of the alternate model against the benchmark.

10 Here we note that the model comparisons in our study are for long-run variance and total variance as provided by the GM models. The comparability tests for volatility predictive ability tests for short run variance of GARCH type models have already been studied extensively, see, Hansen and Lunde (2005) and Gonzalez, Lee and Mishra (2004) and others, and therefore we refrain from reporting that evidence to conserve space.

11 As mentioned earlier, initial work on GM modelling has used various pairwise tests e.g., Diebold-Mariano tests. Recent work on GM modelling and forecasting is transcending to evaluate its gains using stronger statistical tests. For example, Lindblad (2017) relies on the model confidence set approach of Hansen et al. (2011). Similarly, Conrad and Kleen (2020) also use MCS test for GM models. However, both the studies use only the US data.

12 The POS forecasting comparisons are one step ahead only and can be taken as equivalent to a fixed forecasting scheme, while our OS joint tests give multiple steps ahead forecasts. More factually one year at time, for details see section 5.4.

13 The estimates for secular volatility can be obtained through several economic, financial and sentiment related variables.

14 Note that the above convergence only holds for large i, more details can be found in Conrad and Loch (2015) and Engle, Ghysels, and Sohn (2013).

15 In our baseline specification, we use RW-RV in the MIDAS smoothing at daily frequency: each daily realised variance is the rolling sum of 22-daily squared returns. Whereas the long-run variance smoothing for all other GM specifications including macro variables and/or principal components is at monthly frequencies only: long-run variance component changes at monthly frequency and stays constant for the days in a month.

16 The POS, which could also be regarded as in-sample fitting, is a terminology that follows from Engle, Ghysels, and Sohn (2013) and is adopted for ease of comparison.

17 We use innovations after fitting a best fit ARMA model as given by Bayesian information criterion.

18 The exchange rate for France, Germany and the UK are taken against USD, whereas for the US we take it against a basket of currencies, as provided by FRED.

19 The issues pertaining to convergence for MIDAS regressions were frequented quite often due to non-convex objective function when we employed flexible weighting in search of optimal weight structure to exploit the information content in each aggregate variable.

20 PCA analysis has the benefits of over parametrization in model estimations when conditional variance can be influenced by a range of factors and effectively removes noise from the signal. We apply the dynamic PCA such that the stationary macro series are transformed to have standard normal distribution with zero mean and unit variance. We note that the first two components from dynamic PCA invariable explain 70-90% of the variability in the total factor variance of the macro environment of economies investigated in our work.

21 All Appendices are provided in the Supplemental online material file.

22 This choice of a 5-year period is to alleviate potential structural breaks that are possible due to the changing patterns in the run to introduction of the Euro, the Global Financial Crisis and the COVID-19 pandemic. These breaks may affect the estimation of conditional volatilities, both total and long-term, when period variations in different macro and market variables may influence their evolution patterns.

23 Different lag structures are implemented in the estimation process and based on optimal convergence of the model together with minimum use of data for smoothing, 5-year lag (K = 5) is selected.

24 We also compare the MSE forecasting errors of competing errors using another benchmark where we fix the RV in a month i.e. a fixed span GM-RV model. The results of these comparisons are available upon request and qualitatively resemble what we report using the benchmark GM-RV model in this study.

25 The GM estimations show that, across all models and markets, the estimates in the GARCH model are usually significant at 5% or below critical t-values and estimate values are in line to the vast available evidence for GARCH (1,1) models. The regression coefficients, in all models and across markets, on RV in the MIDAS regressions are positive and super significant regardless of the choice of weighting scheme. However, we note that MIDAS input variables in the competing GM models are more often significant with unrestricted weighting scheme at conventional 5% critical t-values. The significance of these estimates with the restricted weighting scheme reduces drastically. These results are not reported to conserve space for the large number of regressions, with even larger number of regression estimates, carried in our work and available upon request.

26 Both are tests for superior predictive ability as ruled by the null hypotheses of RC/White test and SPA/Hansen test.

27 Our results remain unchanged if we increase the number of bootstrap samples to 10,000. Furthermore, we assess the sensitivity of the SPA results with different values for bootstrap parameter ‘q’ that controls for time dependence: a q = 1 completely ignores the time dependence. The results using lower time dependence than q = 0.25 do not alter our results in any manner. We also note that Hansen (2005) and González-Rivera, Lee, and Mishra (2004) also use time dependence values of q = 0.25 that accounts for substantial time dependence corresponding to the empirical observations in the time series modelling of the equity volatility.

28 White (2000) reports that the difference between the naïve p-value and RC test can be described as the data snooping bias in the specification search of better models than the benchmark model.

29 Hansen (2005) shows that RC tests can be manipulated in the presence of poor and irrelevant alternatives in the sample of models. This results in less power and non-rejection of the null hypothesis. Hansen (2005) alleviated this problem in his version of the SPA test by invoking a sample-dependent distribution under the null hypothesis.

30 Multiple steps ahead forecasts for long periods such as five years are not feasible in the context of the models examined in our work, knowing latent variance is time varying and the expected future value of the macroeconomic growth and volatility is not measurable given the time of forecast information filter.

31 Results for POS forecast are available upon request and they largely follow the same pattern that we observe for OS forecasting comparisons.

32 The ${\sigma }_{w1,w2}^2$ and the ${\tau }_{w1,w2}$ refers to the testing results of the forecasting errors of daily total and monthly long run volatilities when the weights for the beta polynomial for each input variable in the MIDAS filter are estimated, while ${\sigma }_{w2}^2$ and ${\tau }_{,w2}$ refers to the testing results when w₁=1 and w₂ is estimated for the beta polynomial of each input variable in the MIDAS filter, respectively.

Additional information

Funding

The corresponding author acknowledge financial support from the project ‘Models for macro and financial economics after the financial crisis’ and supported by Jan Wallanders och Tom Hedelius Stiftelse samt Tore Browaldhs Stiftelse [grant number P18-0201].

Notes on contributors

Nader Virk

Dr. Nader Virk completed his D.Sc. in Finance from Hanken School of Economics, Finland and he is currently a reader in finance at Huddersfield Business School, University of Huddersfield, UK.

Farrukh Javed

Dr. Farrukh Javed has a Ph.D. in Statistics from Lund University, Sweden and he is currently working as Senior lecturer at Department of Statistics, Lund University, Sweden.

Basel Awartani

Dr. Basel Awartani obtained his Ph.D. in Economics, University of London (Queen Mary College), UK and he is currently an Associate Professor at Department of Accounting and Finance, KFUPM, KSA.

Stuart Hyde

Professor Stuart Hyde obtained his Ph.D. in Economics from University of Newcastle and he is Professor of Finance at Alliance Manchester Business School, UK.