In this study, the researcher would like to firstly establish a basic benchmark using the traditional time series model VAR. Moreover, machine learning models SVR, Random Forest, and LSTM are also applied.

VAR model is the multivariant form extended from the Autoregressive Model (AR). An AR model describes how a time series r_t can be regressed to its lag orders. For a multivariate time-series vector X_t, where a_t is the white noise vector, the VAR (p) can be expressed as:

X t = ϕ 0 + ϕ 1 X t − 1 + ϕ 2 X t − 2 + ⋯ + a t

It is worth noting that before building the VAR model, we should ensure that all components of X_t are stationary, which can be tested by ADF tests. After building the VAR model, Granger Causality tests can be applied to check whether each component has granger causality on other components (Tsay, 2005) .

The basic idea of SVR model is that we ensure the point farthest from the hyperplane has a distance d smaller than the tolerant deviation ε, which can be

expressed as a constraint optimization. For the support vector regressor, minimizing distance d can be transformed to the goal of minimizing 1 2 ‖ ω ‖ 2 . For a

linear function y = ω x + b , the constraint becomes: | y i − ( w x i + b ) | ≤ ε , ∀ i . Improvements including introducing slack variables ξ i to the problem or using kernels for mapping are proposed by researchers later (Smola & Schölkopf, 2004; Vapnik, 1999) .

Random Forest is a bagging (bootstrap aggregating) ensemble learning model made up of multiple decision trees, where each decision tree is an estimator of the ensemble learning. In ensemble learning, predictions of each estimator are integrated with other estimators to form the output. Bootstrapping is a sampling-with-replacement method: by randomly selecting n samples each time for k times, we can get k datasets with a sample size of n. Figure 1 shows the basis structure of a random forest (Chu & Qureshi, 2022; Li et al., 2013) .

As a type of recurrent neural network (RNN) model, LSTM is a machine learning method that is adept at processing time series data. However, a traditional RNN model may cause gradient vanishing problem, making the model unable to transfer long term memories. LSTM has been proposed as an improvement to traditional RNN, since it can store long-term memory by adding a memory unit and using gates to control memory storing and discarding. The three gates of a LSTM model are input gate, output gate and forget gate. Besides a hidden state h_t, there is also a memory cell state C_t in the LSTM model. The

state of cell C_t is updated with new information x_t and information passed from h_t−1. The model calculates the information and decides the activation of the gates. Figure 2 shows the basic structure of a LSTM cell. Figure 3 shows the comparisons of pros and cons among the four models (Nosratabadi, Mosavi, Duan, Ghamisi, Filip, Band, Reuter, Gama, & Gandomi, 2020; Staudemeyer & Morris, 2019) .

In order to make accurate forecasts, it is crucial to use input variables which are indicative of economic performance. In the study conducted by Taylan et al., they use economic factors such as GDP, government spendings and productions, and medical data of the pandemic including new cases, vaccinated population, available hospital bed number. In Levanon’s study, he examined three groups of indicators: economic activities, including factors like unemployment and part-time working due to economic reasons; sentimental indicators, including factors like NFIB optimism index and change in consumer comfortable index posted by media; financial indicators, including interest rate swap and LIBOR-Treasury spread (Taylan, Alkabaa, & Yılmaz, 2022; Levanon, 2011) .

In this study, economic and financial factors are desired to be the input of the four models by the researcher. These variables are derived from different aspects, which are summarized by the researcher as People’s Livelihood and Labor Market, Financial Market, International Trade, Inflation and Population. Figure 4 shows the specific variables that are included in this study. By incorporating these variables into the models, we can gain a more comprehensive understanding of the economic landscape and make more accurate predictions about future GDP trends. Other non-economic factors, such as people’s mobility data, and COVID-19 related situation, are discussed in the following part as a supplement of the economic forecasting models.

US has gone through waves of outbreaks in 2020. Figure 5 shows new cases of inflections of US in 2020 based on data provided by WHO². As is shown in the figure, the first massive outbreak in US happened in March 2020. The second outbreak happened in early summer and gradually declined. This may be due to stricter prevention measures. However, the situation worsened again in the fourth quarter of 2020 and continued a rising trend until early January 2021³.

The COVID-19 pandemic caused an unprecedented fall in economic activities in US. According to Federal Reserve’s Monetary Policy Report, over the first half of 2020, the level of US GDP fell a cumulative ten percent and unemployment rate even rose to a post-War high level. The inflation decreased, and the personal consumption expenditures (PCE) also decreased drastically. Financial market was also negatively impacted. The heightened uncertainty and weak demand led many businesses to delay investment plans. In addition, loans became unavailable for firms and consumer borrowings declined since the spending stagnated. Fortunately, as the situation gradually became under control, economic activities recovered gradually, and unemployment rate fell. Although the recover trend slowed down in the late autumn and early winter 2020 with the worsening of the pandemic, the economy resumed. Figure 6 describes how the sudden hit by the pandemic influences important economic variables GDP, Unemployment Rate, PCE, and SP-500 index. The x-axis stands for dates while the y-axis represents the values of these variables. Start from the first quarter of 2020, there was a sudden decline of GDP and PCE and a dramatic increase in unemployment rate. The stock index also experienced a drop. One piece of good news is that GDP and PCE recovered to previous level by the third quarter of 2020, and the stock price continues to rise. The data are retrieved from FRED⁴ (Board of Governors of the Federal Reserve System, 2020, 2021) .

The US government introduced public laws to tackle with health, economic, and people’s livelihood issues brought by the pandemic duly. In March 2020, when the nation was firstly struck by the pandemic, CORONAVIRUS PREPAREDNESS AND RESPONSE SUPPLEMENTAL APPROPRIATIONS ACT⁵ and FAMILIES FIRST CORONAVIRUS RESPONSE ACT⁶ were announced for additional support for CDC, Public Health & Social Services Emergency Fund, and other relevant institutions and funds. The acts aided diagnostics and therapeutics of the virus, aided vaccine development, and provided social assistance like food and insurances. CARES Act⁷ intensively provided details of protection for laborers, assistance or compensation for families and businesses. It supported American’s health care system and fought against the disease. More acts including those aim at extending the supports and stabilizing financial market were announced throughout the battles with the pandemic. Figure 7 presents the timeline of some of the laws announced by the US government during the pandemic.

Google COVID-19 Community Report⁸ shows changes in people’s visits to different categories of places. The data include items of Retail and Recreation, Grocery and Pharmacy, Park, Transit Station, Workplaces, and Residential. The data are collected from 2020 February to 2020 December daily, and are all observed in the form of percentage change from baseline, showing changes of these factors over time.

Figure 8 depicts the trend graph of the data, the x-axis stands for the date, while the y-axis stands for the values of the observed variables. According to the graph, except for “residential change”, other factors went through a fierce drop in March, when US first combated with COVID-19, and “residential change” increased to a positive rate then. Since “residential” measures the duration people stay in residences, it is explicable that instead of visiting other places, people spent more time isolating at home. After the drop, factors except for “residential change” began to recover since the early April. This might be attributed to acts passed by the government since early March, and the nation built up a defense system to better react to COVID-19. The recovery lasted about three months. Compared to the previous level, “parks change” soon increased drastically about two to four times. This might be caused by restricted indoor activities, and people turned to parks to soothe their discontents caused by COVID-19. Other factors that had dropped, including “transit station change”, “retail and recreation change”, “grocery and pharmacy change”, and “workplace change” remained at a negative rate and did not recover to the previous positive level in 2020. What’s more, the rates fluctuated fiercely twice in November and December, which agreed with the new waves of infections in the second half of year 2020. It took less and less time to dissipate each shock, probably because people were more experienced in dealing with the problems.

All the data are retrieved from FRED, Federal Reserve Bank of St. Louis⁹, except for the data of SP-500 P/E Ratio, which are retrieved from Macrotrends¹⁰. The data are collected on a quarterly basis, from the first quarter of 1980(1980-Q1) to the third quarter of 2021 (2021-Q3). The original data are transformed to log differences. The researcher uses the absolute number of the Net Export data, since the original data are always negative. Figure 9 shows how these economic indicators change over the forty years, with the y-axis representing the logarithmic form of the processed data. Table 1 provides an overview of the original data.

Quarterly data spanning from 1980 to 2019-Q4 are used to construct models. The data are used in log difference forms. Instead of a random train-test split, the researcher uses a sequential split on the dataset with a 0.9:0.1 ratio to divide a validation set for the time series data. To construct the VAR model, ADF tests are applied to drop the factors with non-stationary time series. Factors “Population”, “House Price” and “Government Debt” are excluded from the model based on the ADF tests. All factors, except for “Treasury Yield”, are used to construct the SVR model. Additionally, the data are normalized when they are used to build the SVR and LSTM. To make optimal performance of the machine learning models, the researcher uses simple alterations in parameters and Grid Search to tune the parameters. The parameters used in the models are provided in the appendix.

The study would like to find out how the four models VAR(2), SVR, RF, and LSTM perform in the predictions. To evaluate their performance, the researcher uses two common metrics, MSE and R², which provide a general idea of how closely the predictions match the actual values. The metrics are calculated based on log differential data. On the training set, Random Forest outperforms the other three models with a large R² of about 0.77 and a small MSE about 1.18e⁻⁶. Among the remaining models, LSTM performs better than VAR and SVR with a R² of about 0.59. However, it turns out that none of the models performs well in future predictions since R² on testing set are negative. Despite of this, the traditional time series model VAR(2) shows relatively better performance among the remaining models (Table 2).

The DM test is widely used to compare the performance of different models on time series data. The researcher applies the DM test on all training and testing set predictions. The results given by Table 3 reveal that RF outperforms VAR, which is evidenced by a negative DM-statistic and a p-value smaller than 0.05. On the other hand, the performance of SVR and LSTM is comparable to that of VAR and RF. Since training set predictions make up of a major proportion of total predictions, the results from the DM test are consistent with the results given by the evaluation metrics.

Among all the predictions, the primary interest of the study lies in examining forecasts of economic recessions. To this end, the study firstly makes in-sample predictions of log GDP during the US financial crisis from 2008 to 2009. Figure 10 shows the predictions given out by the four models and the true GDP. Also, the study presents the out-of-sample predictions from 2020-Q1 to 2021-Q2, when the country was hit by the COVID-19 pandemic. The predictions are compared with the true log GDP and among all the other models, which are shown in Figure 11. Table 4 provides the out-of-sample predictions of log GDP using the four models.

In both in-sample and out-of-sample predictions, it is evident that the predictions can reflect a rough trend of the data. Nevertheless, during the COVID-19 pandemic, the out-of-sample predictions fall short as all models fail to reflect the violent decline caused by the sudden shock of the pandemic outbreak, and the time series-based model VAR(2) and LSTM are ineffective in showing signs of shocks, as demonstrated by the graph. This could be attributed to the fact that all the models are trained on mostly regular data, making them relatively “obtuse” in this extreme situation. Additionally, the time-series-based model VAR and LSTM are heavily reliant on historical data, which restricts their ability to predict the sudden decline, since the economy was relatively stable and growing steadily before the COVID-19 shock.

The data for the study of Italy are retrieved from Eurostat¹¹, Bank of Italy Statistical Database¹², Wind Database¹³, OECD¹⁴, and FRED, Federal Reserve Bank of St. Louis¹⁵. Table 5 shows the variables that are used in this study.

In order to make reasonable comparisons, resembling variables are chosen for the studies of the two cases. The data of US are available for a longer period of forty years, whereas the data of Italy are from 1995 (1995-Q1) to 2021(2021-Q3) on a quarterly basis. The original data are processed before they are fit into the models. It is worth noticing that the value of variable BOP is converted to BOP + 30,000, and the value of variable Treasury Yield is converted to Treasury Yield + 2. The data are further transformed to log differences, and missing values are filled to ensure completeness of the dataset. Figure 12 shows how these economic indicators change over these years. The y-axis stands for the logarithmic form of the processed data. Table 5 provides an overview of the original data.

Quarterly data spanning from 1995-Q4 to 2019-Q4 are used to build models and make in-sample predictions. The researcher uses a sequential split on dataset with a ratio of 0.9:0.1 instead of a random train-test split. The hyper-parameters of models are displayed in the following table. To construct the VAR model, ADF tests are applied to drop the factors with non-stationary time series. Factors “Population”, “House Price” and “Government Debt” are excluded from the model according to the ADF test. All factors, except for “Treasury Yield”, are used to construct the SVR model. Additionally, the data are normalized when they are used to build the SVR and LSTM.

The evaluation metrics MSE and R², as well as the DM-test, reveal similar results in both the US and Italy cases. According to the evaluation metrics results displayed in Table 6, among the four models, Random Forest outperforms the others on the training set, while VAR(2) is the only model with a positive R² on the testing set. The DM test results shown in Table 7 indicate that the RF model outperforms the VAR, whereas no significant difference is observed between the other models, which aligns with the results concluded from the training set. The comparisons of performance among models in the case of Italy are consistent with those in the case of the US, suggesting that in the time-series predictions, the machine learning models, even the LSTM, which is based on time series, should not be considered superior to traditional VAR in sequence time series predictions.

Like the US case, the researcher makes in-sample predictions of log GDP during the Euro Crisis and presents out-of-sample predictions from 2020-Q1 to 2021-Q2, which cover the period when the pandemic took place. Figure 13 and Figure 14 show respectively the in-sample predictions and out-of-sample predictions generated by the models. Table 8 provides the out-of-sample predictions of log GDP using the four models.

The performance of model prediction resembles that of the US case. The results in both cases indicate that the models can show the trend of how the economy changes, although predictions of the four models deviate from the true values. However, in the case of Italy, the models can’t signal the drastic decline in 2020-Q2 in the out-of-sample prediction clearly.