Forecasting stock indices with the COVID-19 infection rate as an exogenous variable

Average method

The forecasts of all future values are equal to the average (or “mean”) of the data at hand. If we let the existing data be denoted by Y₁, Y₂, …, Y_T, then for forecast horizon h, forecasts for Y_T+h are given by (4)${\displaystyle {\hat{Y}}_{T+h|T}=\frac{\sum _{t=1}^T{Y}_t}{T}=\bar{Y},h\in \mathbb{Z}.}$

Here, h is an integer such that h ≥ 1.

Naïve method

In the Naïve forecast, for any forecast horizon h, the forecast value will be the last observed value in the series. (5)${\displaystyle {\hat{Y}}_{T+h|T}=Y_T,h\in \mathbb{Z}.}$

This method dominates other methods in many situations in economic and financial time series. Since the forecast from a naïve approach is optimal when data follow a random walk, this method is also known as the random walk forecast method.

Drift method

An alternative to the naïve method is to allow the forecasts to increase or decrease over time, where the amount of change over time (also known as drift) is set to be the average change in the data at hand. Thus, the forecast for horizon h is ${\hat{Y}}_{T+h|T}$ and is given by: (6)${\displaystyle {\hat{Y}}_{T+h|T}=Y_T+\frac{h}{T-1}\sum _{t=2}^T\left(Y_t-Y_{t-1}\right)=Y_T+h\left(\frac{Y_T-Y_1}{T-1}\right).}$

This method is equivalent to drawing a line between the first and last observations in the series and extrapolating it into the future.

Autocorrelation function (ACF)

The autocorrection function (ACF) test is a statistical method used to determine the presence of autocorrelation in a time series data set. It measures the correlation between a series and its lags, i.e., the correlation between the data points separated by a given lag interval (Venables & Ripley, 2002). The mathematical formula for ACF is as follows: ${\displaystyle ACF\left(l\right)=\frac{1}{n}\sum _{t=1}^n\frac{\left(Y_t-\bar{Y}\right)\left(Y_{t-l}-\bar{Y}\right)}{S^2}}$ where l is the lag interval; n is the number of observations in the time series; Y_t is the value of the time series at time t; $\bar{Y}$ is the mean of the time series; and S² is the variance of the time series.

The ACF test is used to determine whether a time series is stationary or not. If the autocorrelation is zero or close to zero, the time series is stationary, and the ACF plot will resemble white noise. However, if the autocorrelation is high, then the time series is non-stationary, and the ACF plot will show a pattern of spikes or waves (Box, Genkins & Reinsel, 2015; Brockwell & Davis, 2002). The ACF test is widely used in econometrics, finance, and other fields to analyze time series data. It is a valuable tool for detecting trends, seasonal patterns, and other types of time series behavior (Box, Genkins & Reinsel, 2015).

The Ljung–Box test

The Ljung–Box test is a standard method for model selection and is often used in time series analysis. The Ljung–Box test examines whether there is significant evidence for non-zero correlations at given lags, with the null hypothesis of independence or stationarity in a given time series (Harvey, 1993; Ljung & Box, 1978; Box & Pierce, 1970; Brockwell & Davis, 2002). The Ljung–Box test statistic is calculated as follows: ${\displaystyle Q\left(k\right)=n\left(n+2\right)\sum _l\frac{r_l^2}{n-l}}$ where n is the sample size, k is the number of lags to consider, r_l is the ACF at lag l, and Q(k) is the test statistic which follows chi-squared distribution with k degrees of freedom. A low p-value (e.g., p < 0.10 or 0.05) will indicate the non-stationarity of the series.

Augmented Dickey–Fuller (ADF) test

Another common and familiar statistical method for stationarity in time series literature is the Augmented Dickey–Fuller (ADF) test used to test for the presence of a unit root in time series data (Banerjee et al., 1993; Said & Dickey, 1984). A unit root is a feature of a time series that indicates the presence of a stochastic trend. The ADF test helps determine if a time series is stationary or non-stationary. The mathematical formula for the ADF test is as follows: ${\displaystyle \Delta Y_t=\rho Y_{t-1}+{\delta }_t+{ϵ}_t,}$ where ΔY_t is the first difference of the time series data; ρ is the coefficient of the lagged dependent variable; δ_t is a constant term that includes any deterministic trends in the data, and ϵ_t is the error term. The null hypothesis of the ADF test is that the time series has a unit root, meaning it is non-stationary (Dickey & Fuller, 1979). The alternative hypothesis is that the time series is stationary. The ADF test statistic is compared to a critical value based on the significance level and the sample size. If the test statistic is less than the critical value, the null hypothesis is rejected, and the time series is considered stationary.

The ADF test is commonly used in time series analysis to evaluate the stationarity of a time series and to determine the order of differencing required to make the time series stationary (Stock & Watson, 1993). If the time series is found to be non-stationary, it may be necessary to take first differences or higher order differences to make the time series stationary.

Kwiatkowski-Phillips–Schmidt–Shin (KPSS) test

The KPSS test (Kwiatkowski & Phillips, 1992) is a statistical method used to test for the presence of a unit root in time series data. Unlike the ADF test, the KPSS test assumes that the null hypothesis is stationarity and the alternative hypothesis is non-stationarity. The mathematical formula for the KPSS test is as follows: ${\displaystyle Y_t={\mu }_t+{ϵ}_t}$ where Y_t is the time series data; μ_t is the deterministic trend function; and ϵ_t is the error term. The null hypothesis of the KPSS test is that the time series is stationary, and the alternative hypothesis is that it is non-stationary. The test statistic is calculated based on the sum of squared deviations from the estimated trend function. If the test statistic exceeds the critical value, the null hypothesis is rejected, and the time series is considered non-stationary. The KPSS test is commonly used in time series analysis to evaluate the stationarity of a time series and to determine if differencing is required to make the time series stationary.

For S & P 500 data

For the original series, ACF is not decaying fast for different time lags, so the series is visually non-stationary (see Fig. 4). In contrast, for the first difference of the series, ACF decays very quickly, which is indicative of the stationarity of the differenced series. Further, we must perform a statistical hypothesis test to substantiate the stationarity. We have performed the quantitative tests for testing stationarity by the Ljung–Box test. For the original series (S & P data), p-value < 2.2 × 10⁻¹⁶; and for the first difference series, the p-value is 0.4127. Thus, though the original series is a non-stationary series, the first difference series is stationary by the Ljung–Box test, and these outcomes are consistent with what we have seen from ACF plots. For the original series, the p-valu e < 0.01; for the first difference series, the p-value is 0.01967 from the ADF test. Thus, both the original series and the first difference series do not have unit roots. That is, both the original and the first difference series are stationary by the ADF test. KPSS test provides the p-value of 0.0642 for the original series, and that for the first difference series is greater than 0.10. The original series is not a trend stationary series, but the first difference series is indeed a trend stationary series by quantitative statistical hypotheses tests. Overall, we conclude that the original series of weekly S & P 500 stock indices are not stationary, but the first differences considered here are stationary.

MSCI data

ACF is not decaying fast for different time lags for the original series, so the series is visually non-stationary (see Fig. 5). For the first difference of the series, ACF decays very quickly, which indicates stationarity, but we need further statistical tests to confirm stationarity. For the original series, the Box-Ljung test provides a p-value of < 2.2 × 10⁻¹⁶; but for the first difference series, the p-value is 0.9605. Based on the results presented here, the original series is non-stationary, but the first difference series is stationary. For the original series, the p-value is 0.06524; for the first difference series, the p-value is 0.02559 from the ADF test. Thus, with the smaller nominal significance level (α = 0.05), we may conclude that the first difference series for MSCI is stationary. Therefore, the original series seems to have unit roots, but the first difference series does not. For the original series, the p-value is 0.03821; for the first difference series, the p-value i s > 0.1 from the KPSS test. The difference series seems trend stationary, but the original series was not. Overall, we may conclude that the original series of MSCI stock indices considered here is not stationary, but the first difference of the series is found to be stationary.

VIX data

ACF is not decaying for different time lags for the original series, so the series is visually non-stationary (see Fig. 6). In contrast, for the first difference of the series, ACF decays relatively faster, which is indicative of stationarity. Further, we must perform a statistical hypothesis test to substantiate the stationarity. For the original series, the p-value is 7.737 × 10⁻¹⁶; for the first difference series, the p-value is 0.1808 from the Box-Ljung test. Consequently, though the original series is non-stationary, the first difference series is stationary. For the original series, the p-value is less than 0.01; for the first difference series, the p-value is 0.02095 from the ADF test. Hence, both the original and the first difference series are likely to be stationary. Again, p-values are more significant than 0.01 from the KPSS test for the original and first difference series. Both series are trend stationary. Overall, we may conclude that the original series of the VIX index seems to be trend stationary, but the first difference of the series is undoubtedly stationary.

S & P 500 data

For the original series, the p-value is 1.949 × 10⁻¹⁰ indicating that the original series is not a random walk. Likewise, for the first difference series, the p-value is 0.09555, meaning that the first difference series is also not a random walk if the nominal significance level is 0.10. Thus, in this research, we can employ Box-Jenkin’s methodology for the prediction of S & P 500 stock indices using the first difference series.

MSCI data

For the original series, the p-value is 8.692 ×10⁻¹⁰ indicating that the original series is not a random walk. In contrast, for the first difference series, the p-value of 0.4891 leaves the trace that the first difference series is a random walk.

VIX data

For the original series, the p-value is 5.915 × 10⁻⁰⁹ indicates that the original series is not a random walk. Similarly, for the first difference series, the p-value of 0.04727 demonstrates that the first difference series is also not a random walk at a nominal significance level of 0.05.

However, when a time series is non-stationary, the general practice is to make the series difference stationary. Moreover, if the difference stationary series is not autocorrelated, the original series is a random walk. If so, any parametric time series modeling should be used for forecast purposes. In our preliminary analysis, we have found that the first difference series of S & P and VIX are stationary and autocorrelated (not random walk). Still, the first difference series of MSCI is a stationary but random walk. Thus, we may employ a parametric method for price or index forecasting for the first difference between S & P 500 series and VIX, which is Box-Jenkin’s methodology. On the other hand, we may deploy random walk forecasting methods for MSCI index forecasting.

S & P 500 weekly indices and weekly COVID-19 cases

Here, we have considered weekly data for both S & P 500 index and COVID-19 cases in the USA. The Pearsonian product-moment correlation between S & P 500 weekly index and Weekly COVID-19 cases in the USA is 0.5500 with a p-value of 6.204 10⁻⁰⁵. Therefore, the number of confirmed COVID-19 cases is significantly correlated with S & P 500 stock indices.

Weekly VIX and weekly COVID-19 cases

Here, we have considered weekly data for VIX and COVID-19 cases in the USA. The Pearsonian product-moment correlation between VIX and Weekly COVID-19 cases in the USA is −0.2487 with a p-value of 0.09184. Therefore, the number of confirmed COVID-19 cases significantly correlates with VIX at a nominal significance level of 0.10.

S & P 500 index data

In this study, we have considered the weekly number of COVID-19 confirmed cases as a regressor and weekly S & P 500 indices from December 12, 2020, to February 12, 2021 (9 weeks) for forecasting using our optimum trained model. Forecast indices and 80%, 95%, and 99% Prediction intervals are presented in the following table (see Table 4). The visualization of these results has been presented in Fig. 8.

VIX data

Similar to S & P 500, we have considered the weekly number of COVID-19 confirmed cases as a predictor variable and weekly VIX data from December 12, 2020, to February 12, 2021 (9 weeks) for forecasting using our optimum trained model. Forecast indices, along with 80%, 95%, and 99% prediction intervals, are presented in the following table (see Table 5). These results have been displayed in Fig. 9 along with the original series. Since VIX is an index and it cannot take a negative value. To address this issue, we fit the model on natural logarithm-transformed data and, later on, exponentiated the results to bring them back to their original scale.

MSCI data

Like S & P 500 Index, we have considered the weekly MSCI Index from December 12, 2020, to February 12, 2021 (9 weeks) for forecasting using our train model. Here, forecasts have been made using three different random walk forecasting methods: the mean method, the naïve method, and the drift method, as described in the fourth section. Forecast of the index along with 80%, 95%, and 99% Prediction intervals are presented in the following tables (see Tables 6, 7, and 8) for the aforementioned methods. These results have been shown schematically in Fig. 10, along with the original series.

From the accuracy measures (Hyndman & Athanasopoulos, 2018; Hyndman & Koehler, 2006; Armstrong, 1978) presented in Table 9, it can be concluded that the best method for MSCI data forecasting, based on the RMSE and MPAE, is the drift method, which suggests that the trend is more important than the seasonality in this series.