Since the seminal work of Ghysels (2004), a proliferation of literature has demonstrated the usefulness of Mixed data sampling (MIDAS) modelling. Given the non-linear nature of MIDAS, most models only contain one predictor. As a result, combinations of MIDAS models with or without traditional single‑interval models (e.g. ARMA, VAR) are common. When point forecasting, combining direct MIDAS model forecasts with iterative single-interval model forecasts is straightforward. However, the calculation of prediction intervals around the point forecasts is less straightforward, and to our knowledge has not been addressed in the literature.
It is well understood that model combinations are an effective way to deal with model uncertainty, and that equally weighted model averages generally outperform more sophisticated weighting schemes (Elliot and Timmerman 2004). Most of the forecast combination literature has focused on point forecasts, with far less attention devoted to prediction interval estimation. Analytical expressions for single-interval model prediction intervals are well established, and so is the bootstrap as a means of combining models that employ iterated or direct forecasts. To our knowledge we are the first paper to consider prediction interval estimation for a model average that combines iterated and direct forecasts.
For illustrative purposes we forecast Australian household consumption (henceforth, consumption). Consumption is of interest to policymakers and academics as it’s a direct measure of living standards and the largest component of Gross Domestic Product (GDP). Contributing approximately 70 per cent of GDP, small changes in consumption can have a larger impact on GDP than any other contributing variable. This is important because consumption can be sensitive to economic conditions, with households quick to increase precautionary savings amidst uncertainty. Goods and services taxes are also a large source of government revenue, so accurate forecasts are needed to inform policy.
Macro-economic variables are typically published on a quarterly or monthly frequency, even though contractions can occur within weeks. Traditional econometric approaches that convert high-frequency data to the lowest common frequency compromise forecast timeliness and suffer from information loss and parameter bias (Ghysels et al. 2004).1 The MIDAS model avoids these issues by regressing low-frequency data against high frequency predictors. Duarte et al. (2017) for example use daily data with other low-frequency economic variables to forecast Portuguese consumption, and Morita (2022) uses daily stock returns to forecast Japanese GDP growth. Despite the ubiquity of MIDAS, limited research has applied the model class to the Australian macro-economy.
Our consumption data commences in the 1st quarter (Q1) 1959 and ends Q1 2022. We consider one‑step ahead forecasts commencing Q2 2003 using rolling windows. We employ a range of MIDAS models and single-interval benchmarks and consider variables related to spending activity, household finances, employment, residential property, inflation, interest rates, market indicators and trade.
Consistent with Verbaan et al. (2017), we find that direct measures of spending provide the most forecasting power. Equally weighted combinations of models with equal predictive ability (EPA), show that during normal economic periods there is no difference between single-interval benchmarks and combinations containing MIDAS forecasts. However, model combinations that include MIDAS models significantly improve forecasts during periods of high uncertainty. Our procedure for estimating PIs around model combinations of direct and iterated forecasts performs as well as standard PIs around benchmarks. All PI interval violation rates are higher than the level of confidence when the out‑of‑sample period includes the GFC and COVID‑19 periods. This is largely due to the inability of our models to identify significant turning points during crises. Over more normal times however, our PIs are consistent with the level of confidence.
The remainder of this paper is structured as follows. Section 2 outlines the MIDAS class of models. Section 3 outlines our bootstrap procedure for prediction interval estimation around model averages that consist of direct and iterated forecasts. Section 4 is an empirical application to Australian household consumption. Section 5 concludes.
Footnotes
[1] The parameter (or discretisation) bias arises because construction of the lower frequency variable imposes a weighting scheme on the data. If for example a monthly average is constructed using daily data, this imposes an equal weight on each of the daily observations. If an alternative weighting scheme like a hump shape is more appropriate, imposing equal weights by constructing an average will introduce bias.
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