Prediction intervals of the COVID-19 cases by HAR models with growth rates and vaccination rates in top eight affected countries: Bootstrap improvement

Abstract

This paper is devoted to modeling and predicting COVID-19 confirmed cases through a multiple linear regression. Especially, prediction intervals of the COVID-19 cases are extensively studied. Due to long-memory feature of the COVID-19 data, a heterogeneous autoregression (HAR) is adopted with Growth rates and Vaccination rates; it is called HAR-G-V model. Top eight affected countries are taken with their daily confirmed cases and vaccination rates. Model criteria results such as root mean square error (RMSE), mean absolute error (MAE), $R^2$ , AIC and BIC are reported in the HAR models with/without the two rates. The HAR-G-V model performs better than other HAR models. Out-of-sample forecasting by the HAR-G-V model is conducted. Forecast accuracy measures such as RMSE, MAE, mean absolute percentage error and root relative square error are computed. Furthermore, three types of prediction intervals are constructed by approximating residuals to normal and Laplace distributions, as well as by employing bootstrap procedure. Empirical coverage probability, average length and mean interval score are evaluated for the three prediction intervals. This work contributes three folds: a novel trial to combine both growth rates and vaccination rates in modeling COVID-19; construction and comparison of three types of prediction intervals; and an attempt to improve coverage probability and mean interval score of prediction intervals via bootstrap technique.

Keywords: Bootstrap procedure; COVID-19; HAR model; Mean interval score; Prediction interval.

Fig. 1

Flow chart for this research of HAR-G-V model.

Fig. 2

Growth rates of daily COVID-19 confirmed cases (7-day smoothed).

Fig. 3

Vaccination rates.

Fig. 4

Sample autocorrelation coefficient function (ACF) of daily COVID-19 confirmed cases (7-day smoothed).

Fig. 5

Daily confirmed cases (7-day smoothed) and HAR-G-V model fitting with its residuals.

Fig. 6

Distribution of residuals by HAR-G-V models.

Fig. 7

One-day ahead predicted values and 95% prediction intervals by HAR-G-V models: Box–Jenkins prediction intervals (left column) and Bootstrap prediction intervals (right column); Brazil, France, Germany and India.

Fig. 8

One-day ahead predicted values and 95% prediction intervals by HAR-G-V models: Box–Jenkins prediction intervals (left column) and Bootstrap prediction intervals (right column); Italy, Russia, UK and USA.

Fig. 9

Coverage probability (left column) and Mean interval score (right column) of 80%, 95% prediction interval (PI) for Brazil.

Fig. 10

Coverage probability (left column) and Mean interval score (right column) of 80%, 95% prediction interval (PI) for India.

Fig. 11

Coverage probability (left column) and Mean interval score (right column) of 80%, 95% prediction interval (PI) for Italy.

Fig. 12

Coverage probability (left column) and Mean interval score (right column) of 80%, 95% prediction interval (PI) for Russia.