Two-Covariate GARCH-MIDAS Volatility Model

• The two-covariate GARCH-MIDAS model is a mixed-frequency framework that decomposes high-frequency return volatility into a short-run GARCH component and a long-run MIDAS component driven by distinct low-frequency covariates.
• It employs additive or log-additive long-run specifications with common or separate MIDAS weighting schemes to capture incremental and non-redundant predictive information from financial and macroeconomic variables.
• Empirical applications in crude-oil and copper studies show the second covariate enhances forecasting only when it provides independent predictive signals, balancing model flexibility with parsimony.

Searching arXiv for recent and foundational papers on two-covariate GARCH-MIDAS and related mixed-frequency volatility models. Searching arXiv for "GARCH-MIDAS two covariate volatility". A two-covariate GARCH-MIDAS model is a mixed-frequency volatility specification in which the conditional variance of high-frequency returns is decomposed into a short-run GARCH-type component and a long-run MIDAS component driven by two low-frequency covariates. In the cited literature, the two covariates are introduced to capture distinct slow-moving determinants of volatility, such as realized volatility together with macroeconomic or uncertainty variables, and they enter the long-run component either additively in levels or additively in log form (Dai et al., 2020, Wang et al., 2024). Closely related multiplicative long-run/short-run models for realized volatility preserve the same structural logic and sharpen the interpretation of normalization, persistence, and weighting in multi-component volatility systems (Amendola et al., 2020).

1. Conceptual position within mixed-frequency volatility modeling

The GARCH-MIDAS framework separates volatility dynamics by frequency. Daily returns are modeled with a high-frequency conditional variance that is factored into a rapidly adjusting component and a slowly evolving component. The short-run term captures volatility clustering at the daily horizon; the MIDAS term aggregates lower-frequency predictors through distributed lags. In this architecture, the two-covariate variant extends the long-run component from a single predictor to a pair of predictors, allowing the model to combine information from different economic channels, such as market-based realized volatility and macro-financial variables (Dai et al., 2020).

In the crude-oil application, the distinction is expressed as a comparison between single-factor models and two-factor models, where the two-factor specification adds a second low-frequency regressor to the long-run variance. In the copper application, the same idea appears as a multi-covariate MIDAS term, with the long-run component written as a function of one macro variable or realized volatility plus another macro variable. Both implementations retain the same underlying decomposition of conditional variance into a short-run GARCH-type term and a long-run MIDAS term (Wang et al., 2024).

A recurrent theme is that the second covariate is not automatically useful. The empirical role of the additional covariate depends on whether it contributes independent predictive information rather than information already absorbed by the first factor. This suggests that two-covariate GARCH-MIDAS is fundamentally a model of incremental long-run variance information, not merely a larger parameterization.

2. Core mathematical specification

A standard return equation in the GARCH-MIDAS class is

$$r_{i,t} = \mu + \sqrt{g_{i,t} T_t}\,\varepsilon_{i,t}, \qquad \varepsilon_{i,t}\mid\mathcal{F}_{i-1,t}\sim N(0,1),$$

so that

$$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$

Here, $i$ indexes trading days within low-frequency period $t$, $g_{i,t}$ is the short-run component, and $T_t$ or $\tau_t$ is the long-run component (Wang et al., 2024).

In the crude-oil specification, the short-run term is a normalized GARCH(1,1),

$$g_{i,t} = (1-\alpha-\beta) + \alpha \frac{(r_{i-1,t}-\mu)^2}{\tau_t} + \beta g_{i-1,t}, \qquad \alpha>0,\ \beta>0,\ \alpha+\beta<1,$$

so that $E[g_{i,t}]=1$ and the slow-moving level of volatility is absorbed by $\tau_t$ (Dai et al., 2020). In the copper specification, the short-run term is instead a mean-reverting, unit-variance GJR-GARCH(1,1),

$$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$0

with the usual positivity and mean-reversion restrictions (Wang et al., 2024).

For the two-covariate long-run component, one common linear-additive specification is

$$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$1

where $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$2 is realized volatility and $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$3 is a macro variable such as the GEPU index or $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$4. A rolling-window version replaces $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$5 by a daily-updated $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$6 and uses rolling-window covariates (Dai et al., 2020).

A second common form is the log-specification,

$$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$7

which implies $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$8. In this setup, each covariate can in principle have its own MIDAS weighting parameters, although restricted weighting schemes are commonly used in practice (Wang et al., 2024).

An important point is that the cited implementations use additive or log-additive long-run terms. The copper paper explicitly notes that there is no multiplicative interaction in $$\operatorname{Var}(r_{i,t}\mid\mathcal{F}_{i-1,t}) = g_{i,t}T_t.$$9; the nonlinearity comes via the exponential if the log-specification is used (Wang et al., 2024).

3. Covariates, transformations, and frequency alignment

The low-frequency covariates used in two-covariate GARCH-MIDAS models span financial and macroeconomic predictors. Realized volatility is defined at the monthly horizon as

$i$0

and serves as a benchmark predictor of long-run volatility (Dai et al., 2020). In rolling-window variants, realized volatility can also be computed over a daily index using a 22-trading-day window, producing a smoother daily-updated long-run component (Dai et al., 2020).

The crude-oil study uses the Global Economic Policy Uncertainty index and its changes. The change series is defined as

$i$1

The distinction between the level and the change is substantive: the level proxies persistent global policy uncertainty, whereas the change reflects shifts or surprises in policy uncertainty. The reported results show that $i$2 is more tightly linked to long-term crude-oil futures volatility than the level GEPU index, especially once realized volatility is included (Dai et al., 2020).

The copper study considers daily and monthly macro variables, including IR, Slope, PPI, IP, PMI, CSI, NOI, NAI, and DI, and also constructs macro volatilities by fitting AR models to macro indices and using squared residuals as proxies for macro volatility. Monthly variables are mapped to the daily sequence by holding the monthly value constant within the month. Daily IR and Slope are used directly, and when they enter the MIDAS term with large lag lengths they are treated as low-frequency relative to daily returns (Wang et al., 2024).

These implementations imply two broad design patterns. One pattern pairs a financial volatility proxy with a macro or uncertainty covariate, as in $i$3. The other pairs two macro determinants, as in the copper bivariate macro-macro models. This suggests that the term “two-covariate” is best understood as a statement about the structure of the long-run component rather than the economic type of the regressors.

4. MIDAS weighting, normalization, and parameterization

Two-covariate GARCH-MIDAS models use MIDAS lag weights to compress long distributed lag structures into a small number of shape parameters. A standard Beta-polynomial weighting scheme is

$i$4

or, in normalized $i$5 form,

$i$6

The weights are positive and normalized to sum to one (Dai et al., 2020, Wang et al., 2024).

A common practical restriction is to fix $i$7 and estimate only $i$8. In the crude-oil paper, this follows Engle et al. and Asgharian et al.; in the copper paper, it is termed a restricted weighting scheme and is used to ensure decaying weights and reduce parameter proliferation (Dai et al., 2020, Wang et al., 2024). Under this restriction, more recent lags often receive larger weights, and the oil paper notes that the estimates often imply that more recent values of $i$9 and $t$0 carry more weight in determining current long-run volatility (Dai et al., 2020).

The treatment of weights differs across two-covariate designs. In the crude-oil two-factor model, both covariates use the same MIDAS weight function and therefore share the same lag-decay shape (Dai et al., 2020). In the copper multi-covariate specification, each macro series can in principle have its own weighting parameters and even its own lag length, although restricted weights are mostly used in the empirical work (Wang et al., 2024). The first choice imposes a common memory structure; the second allows heterogeneous memory across covariates.

This difference is not merely technical. A common weighting function imposes that the relevant memory length and decay shape are the same for both predictors. Separate weighting functions permit the two covariates to affect long-run variance over different temporal horizons. This suggests a basic specification trade-off between parsimony and flexibility.

5. Estimation, forecasting, and empirical evidence

The cited studies estimate GARCH-MIDAS models by maximum likelihood or quasi-maximum likelihood under conditional normality. For the crude-oil specification, the daily log-likelihood contribution is

$t$1

with analogous replacement of $t$2 by $t$3 in rolling-window models. Model comparison uses LLF and BIC, while the copper paper also reports the variance ratio $t$4 to measure how much of total conditional variance is explained by the long-run component (Dai et al., 2020, Wang et al., 2024).

In the crude-oil forecasting exercise, the data span is Dec 1998–Oct 2019, with a 13-year calibration window, an additional 3-year pre-sample, and a 5-year out-of-sample period. Forecasts are 1-day ahead conditional variances, evaluated with RMSE and RMAE on variance and RMSD and RMAD on standard deviation. Differences in predictive accuracy are assessed with the Diebold–Mariano statistic, where a negative DM indicates that the row model has lower loss than the column model (Dai et al., 2020).

The empirical findings reported for representative two-covariate implementations are summarized below.

Paper	Two-covariate design	Reported result
(Dai et al., 2020)	$t$5 and $t$6	Rolling-window models outperform fixed-span models; Model X ($t$7) is the best performing specification for both Brent and WTI
(Wang et al., 2024)	RV + macro; macro + macro	Bivariate models have higher log-likelihood, lower BIC, and larger VR than univariate models
(Wang et al., 2024)	PPI-based MIDAS filters	PPI is the most efficient macroeconomic variable

The crude-oil results are especially explicit about incremental information. In single-factor rolling-window models, $t$8 significantly outperforms GEPU, with DM statistics of $t$9 for Brent and $g_{i,t}$0 for WTI. In two-factor rolling-window models, Brent RMSE falls from $g_{i,t}$1 in Model IX ($g_{i,t}$2) to $g_{i,t}$3 in Model X ($g_{i,t}$4), while WTI RMSE falls from $g_{i,t}$5 to $g_{i,t}$6. The paper concludes that GEPU is not an effective forecast factor in the two-factor setting, whereas $g_{i,t}$7 remains significant and becomes the main source of long-term volatility of crude-oil futures once both factors are used (Dai et al., 2020).

The copper study reaches a parallel but more general conclusion. Significant level drivers include IR, IP, and PPI; significant volatility drivers include Slope, PPI, CSI, and DI; and realized volatility is extremely powerful. In bivariate models, adding a second variable improves fit, echoing Conrad and Kleen’s “two are better than one.” PPI stands out as the only variable that remains important across univariate level models, univariate volatility models, bivariate macro-macro models, mixed level-plus-volatility models, GARCH-MIDAS-RV+X models, and DCC-MIDAS for correlations (Wang et al., 2024).

6. Interpretation, misconceptions, limitations, and related extensions

A common misconception is that statistical significance in a single-factor GARCH-MIDAS model guarantees relevance in a two-covariate specification. The crude-oil evidence directly contradicts this. GEPU is significant and positive in single-factor models, but once realized volatility is added, the GEPU coefficient becomes small and not significant for Brent, and negative and marginally significant for WTI, while the realized-volatility term can also lose significance in that specification. The authors interpret this as overlapping information: the level GEPU index largely shares predictive content with realized volatility, whereas $g_{i,t}$8 retains non-redundant information (Dai et al., 2020).

A second misconception is that the second covariate should necessarily be another macro level series. The copper evidence shows a broader design space: the second covariate may be a macro volatility, a second macro level, or realized volatility paired with a macro variable. The strongest combinations are those that are economically distinct and empirically complementary, such as PPI level with Slope volatility, or RV with PPI (Wang et al., 2024).

The main technical limitations emphasized in the literature are collinearity, parameter proliferation, and identification. The copper study notes that several macro series are correlated and that adding a second macro variable can make some coefficients insignificant, indicating that the variables absorb the same predictive content. It also notes that unrestricted weighting schemes in bivariate models are avoided partly to contain parameter proliferation and partly because available software does not easily allow different unrestricted weighting schemes in bivariate models (Wang et al., 2024). The oil study reaches a similar practical resolution by using the same MIDAS weight function for both covariates in the two-factor model (Dai et al., 2020).

The question of how many low-frequency covariates to include remains open, but the copper paper reports that Conrad and Kleen (2019) found little improvement when moving from two to three macro covariates in GARCH-MIDAS. This suggests that two-covariate specifications often represent a pragmatic upper bound before overfitting pressures become more serious (Wang et al., 2024).

Related multiplicative volatility models reinforce the same structural lesson. The DMEM and MEM-MIDAS framework decomposes realized volatility into a slow-moving long-run component and a fast-moving short-run component, with the long-run term written in exponential MIDAS form and the short-run term following a GARCH-like recursion. The paper also discusses DAGM, where positive and negative values of the same covariate enter the long-run component through separate MIDAS filters. That construction is structurally close to a two-term MIDAS long-run component and provides a template for more flexible multi-driver volatility decompositions (Amendola et al., 2020).

Taken together, the cited literature characterizes the two-covariate GARCH-MIDAS model as a parsimonious mixed-frequency variance decomposition in which the long-run component is enriched by a second slow-moving determinant. Its empirical success depends less on the mere presence of two covariates than on whether the second covariate captures an economic dimension or a surprise component not already internalized by the first.