Score-Driven Interactions for “Disease X” Using COVID and Non-COVID Mortality

Abstract

The COVID-19 (coronavirus disease of 2019) pandemic is over; however, the probability of such a pandemic is about 2% in any year. There are international negotiations among almost 200 countries at the World Health Organization (WHO) concerning a global plan to deal with the next pandemic on the scale of COVID-19, known as “Disease X”. We develop a nonlinear panel quasi-vector autoregressive (PQVAR) model for the multivariate t-distribution with dynamic unobserved effects, which can be used for out-of-sample forecasts of causes of death counts in the United States (US) when a new global pandemic starts. We use panel data from the Centers for Disease Control and Prevention (CDC) for the cross section of all states of the United States (US) from March 2020 to September 2022 regarding all death counts of (i) COVID-19 deaths, (ii) deaths that medically may be related to COVID-19, and (iii) the remaining causes of death. We compare the t-PQVAR model with its special cases, the PVAR moving average (PVARMA), and PVAR. The t-PQVAR model provides robust evidence on dynamic interactions among (i), (ii), and (iii). The t-PQVAR model may be used for out-of-sample forecasting purposes at the outbreak of a future “Disease X” pandemic.

1. Introduction

Although the COVID-19 (coronavirus disease of 2019) pandemic is over, the probability of such a pandemic is about 2% in any year, and a pandemic on the scale of COVID-19 could likely be within the next 59 years (Marani et al. 2021). The British Broadcasting Corporation (BBC) explores six diseases most likely to cause the next global pandemic (Constable and Kushner 2021). Peel (2024) reports international negotiations among 194 countries at the World Health Organization (WHO) on the world’s first pandemic treaty, i.e., a global plan to deal with the next global pandemic known as “Disease X”. Motivated by this, we develop a new model that can be used for out-of-sample forecasts of causes of death counts in the United States (US) when a new global pandemic starts.

We use data on COVID-19 and non-COVID-19 deaths in the US from the Centers for Disease Control and Prevention (CDC) for all US states. The literature on the COVID-19 pandemic suggests that the possible causes of COVID-19 mortality may directly or indirectly influence other causes of mortality (Appleby 2020; Cronin and Evans 2021; Rosenfeld et al. 2023). For example, the number of homicides increased by 20.9% in the US from 2019 to 2020 during the first nine months of the year, due largely to gun violence (Barrett 2020; Cronin and Evans 2021). In New York City, the number of homicides increased by nearly 40% and shootings almost doubled from 2019. Some of the largest increases in homicide rates during the COVID-19 pandemic occurred in relatively small cities where the rate of homicide was historically relatively low (Barrett 2020; Rosenfeld et al. 2023). In cities with fewer than 10,000 residents, the number of homicides increased by more than 30% from 2019 to 2020 during the first nine months of the year. Another example is the sharp increase in the number of deaths from uninfected Alzheimer’s and other forms of dementia patients (Wan 2020). From March to September 2020, more than 134,200 people died from Alzheimer’s and other forms of dementia in the US. This is 13,200 more deaths than expected based on the same period of 2019, and many of those are due to panic isolation or no access to home health care (Appleby 2020; Wan 2020).

The main research question addressed is how to measure the dynamic interactions between possible causes of COVID-19 and non-COVID-19 deaths in the US. To answer, we use a new robust econometric model. We extend the statistical models of the literature, and we use a multivariate dynamic panel data model with score-driven individual-specific effects, named the panel quasi-vector autoregressive (PQVAR) model, to measure the dynamic interactions among different possible causes of mortality for all US states and the District of Columbia from March 2020 to September 2022. The PQVAR model is an extension of the QVAR model of Blazsek et al. (2021) to panel data, and it is in the class of dynamic conditional score (DCS) models (Harvey 2013; Harvey and Chakravarty 2008), also named generalized autoregressive score (GAS) models (Creal et al. 2011, 2013), which we call score-driven models.

Using the publicly available data from CDC, we classify the counts of causes of death as (i) due to COVID-19 $y_{1,i,t}$, (ii) death that may be medically related to COVID-19 $y_{2,i,t}$, and (iii) the rest of the causes $y_{3,i,t}$. Category (ii) includes causes of death that can be caused by COVID-19 or existing medical conditions that can severely be influenced by COVID-19. Category (iii) includes causes of death that are indirectly related to COVID-19, for example due to changes in population behavior during the pandemic. Hence, the death counts in categories (i)–(iii) may be positively correlated, and by jointly modeling them, we may improve the forecasts for future global pandemics.

The objective of this paper is to develop a robust dynamic association measure among the dependent variables $y_{i,t}={(y_{1,i,t},y_{2,i,t},y_{3,i,t})}^{\prime }$. We show that (i)–(iii) are significantly positively correlated with each other, i.e., evidence of in-sample forecasting performance. We use death count data instead of mortality rate data (Ionides et al. 2013; Ruhm 2000) because we study the within-state interactions among the components of $y_{i,t}$, and it does not matter whether we divide $y_{i,t}$ by the state population.

Dynamic interactions are measured using the impulse response function (IRF). Estimates may be useful for future pandemics with a similar global impact as COVID-19. We present the following technical details on PQVAR: (i) reduced-form t-PQVAR(p) representation; (ii) first-order and PVMA(∞) representation (panel vector moving average, PVMA); (iii) IRF formulation; (iv) maximum likelihood (ML) method and conditions of its asymptotic properties for PQVAR; (v) Gaussian-PVARMA(p,p) (PVAR moving average) and Gaussian-PVAR(p) are special cases of t-PQVAR(p). The panel data method developed in this paper can be used in different contexts from the COVID-19 pandemic, for example in microeconometrics (e.g., marketing research or human resources).

Our contributions are the following: (i) As we know, we are the first to estimate dynamic interactions between COVID-19 and non-COVID-19 mortality. (ii) We use updated data for COVID-19 concerning other works from the literature. (iii) The PQVAR model may help with out-of-sample forecasts of different causes of death categories for future pandemics. We show the superiority of t-PQVAR(p) with respect to Gaussian-PVARMA(p,p) and Gaussian-PVAR(p) for COVID-19 and non-COVID-19 mortality from 2020 to 2022. The ML conditions are supported for t-PQVAR. We find significantly positive interaction effects of the causes of death of COVID-19 mortality and non-COVID-19 mortality.

In the remainder of this paper, Section 2 reviews the literature, Section 3 presents the data and econometric methods, Section 4 presents the results, Section 5 discusses them, and Section 6 concludes.

2. Literature Review

In this section, we first review the literature concerning possible causes of COVID-19 and non-COVID-19 mortality. Second, we review the literature on score-driven time series models citing some works that use multivariate models, and we also relate PQVAR to the literature on panel data models.

2.1. COVID-19 and Non-COVID-19 Mortality

Appleby (2020), using data from the United Kingdom (UK) until April 2020, suggests that possible causes of non-COVID-19 deaths may increase during the pandemic because of the following points: (i) People may not seek help from the National Health Service of the UK, because they fear contracting COVID-19 or they do not want to burden the National Health Service during the pandemic. (ii) COVID-19 may cause severe pulmonary, respiratory, heart, prostate, or diabetes diseases that may result in death. (iii) Early in the pandemic, COVID-19-originated deaths may have been classified as influenza or pneumonia. For the US, see the similar arguments of Barach et al. (2020).

Cronin and Evans (2021), using data from the US from 2017 to 2020, show that in the US, about 13% of the excess mortality is due to non-COVID-19 causes of death in 2020. According to those authors, many of these causes are due to social distancing to reduce the number of infections and the negative influences of the pandemic on the US economy. For example, the high rates of deaths due to suicide, drug overdose, alcohol-related liver disease, murders, and uninfected patients with Alzheimer’s disease without access to health care indicate that social distancing and the closing of the economy may have increased the number of non-COVID-19 deaths.

Rosenfeld et al. (2023) use US crime rate data from 34 cities during 2020. They find that murder rates increased, i.e., homicide rates in 2020 were 30% higher than in 2019, and domestic violence also increased during the first months of the COVID-19 pandemic. See also Barrett (2020). Woolf et al. (2021, 2020), using COVID-19 data from the US for March–July 2020 and March–October 2020, respectively, find increases in non-COVID-19 deaths. The former paper studies COVID-19 and non-COVID-19 deaths by age groups. The latter paper estimates the determinants of causes of excess deaths and finds that those are only partly explained by COVID-19. Jacobson and Jokela (2020), using data from the US from March to May 2020, focus on excess non-COVID-19 deaths by age and gender groups. They find unexpected deaths in several age and gender groups beyond the COVID-19 deaths for the first three months of the COVID-19 pandemic. Bhaskaran et al. (2021), using data from the UK from February to November 2020, show that mortality due to COVID-19 is linked with age and pre-existing medical conditions. They investigate how specific factors are differently associated with COVID-19 and non-COVID-19 mortality.

Shiels et al. (2021a), using US data from March to August 2020, study excess non-COVID-19 deaths by age groups. They find that most unexpected non-COVID-19 deaths were in April, July, and August of 2020, were for persons aged 25 to 64 years, and were caused by diabetes, Alzheimer’s disease, and heart disease. Shiels et al. (2021b), using US data from March to December 2020, find that 26% of the 2.88 million deaths can be attributed to non-COVID-19 deaths. Other direct possible causes are lung diseases, influenza, pneumonia, heart diseases, prostate diseases, and diabetes, which are possibly caused by or whose existing conditions are severely influenced by COVID-19. See Askin et al. (2020) and Aquino-Matus et al. (2022).

The papers cited in this section do not implement dynamic specifications to measure the interaction effects between COVID-19 and non-COVID-19 possible causes of death. We extend these works because we estimate the dynamic interaction effects between the causes of death corresponding to COVID-19 and non-COVID-19 mortality in addition to the descriptive contemporaneous analysis reported in the literature using a greater dataset from the CDC.

2.2. Score-Driven Models

Score-driven models are observation-driven models that are robust alternatives to Gaussian linear state-space models (Harvey 1990) and classical observation-driven models (e.g., AR moving average, ARMA; generalized AR conditional heteroscedasticity, GARCH; VARMA), with the following statistical advantages: (i) Score-driven models are optimal from an information-theoretic perspective (Blasques et al. 2015). Asymptotically, a score-driven update reduces the distance between the true conditional density and the conditional density implied by the score-driven model locally, in expectation, and at every step, even for misspecified score-driven models. (ii) Score-driven updates also satisfy optimality properties based on a global definition of Kullback–Leibler divergence (Gorgi et al. 2023). Score-driven updates reduce the distance between the expected updated parameter and the pseudo-true parameter, and the optimality result of Blasques et al. (2015) can hold globally over the parameter space. (iii) Score-driven filters are robust to missing data and extreme observations (Harvey 2013). (iv) The updating terms of several classical observation-driven models are special cases of updating terms of score-driven models. For the statistical inference of score-driven models, see Creal et al. (2013), Harvey (2013), and Blasques et al. (2022).

We refer to some works that extend the multivariate score-driven location filter of Harvey (2013). See Blazsek et al. (2021, 2022, 2023, 2024a) on the IRFs for score-driven multivariate location filters, Markov regime-switching score-driven multivariate location filters, cointegrated score-driven location filters, and a combination of multivariate score-driven location and volatility filters, respectively. We also refer to the recent works of Blasques et al. (2024) and Delle Monache et al. (2024). Blasques et al. (2024) show the ML conditions for non-stationary score-driven location models. Delle Monache et al. (2024) model permanent and transitory changes using score-driven location models. We also refer to Blazsek and Escribano (2022, 2023) and Blazsek et al. (2024b), who forecast climate variables using multivariate score-driven filters. Our PQVAR model extends these multivariate time series models to a panel data setup with unobserved effects.

The score-driven updating mechanism of PQVAR extends the linear updating mechanism of the PVAR model (Binder et al. 2005), and it also extends the multivariate score-driven location filter of Harvey (2013) to panel data with dynamic individual-specific effects. We implement the ML theory of Harvey (2013); Creal et al. (2013), and Blasques et al. (2022) to panel data, for which the cross-sectional dimension N is fixed and the time series dimension $T\to \infty$. In this paper, we prefer this choice because N represents all US states and the District of Columbia, i.e., it is fixed.

3. Material and Methods

In this section, we start with a description of the dataset used in this research. Then, we present the PQVAR model, its estimation procedures, special cases (i.e., the PVARMA and PVAR models), and the specification and identification of the IRFs of the PQVAR, PVARMA, and PVAR models.

3.1. Data

The COVID-19 outbreak became a public health emergency of international concern according to the WHO on 30 January 2020, and COVID-19 became a pandemic on 22 March 2020. We use monthly panel data from March 2020 to September 2022 ($T=31$) for 50 states in the US and the District of Columbia ($N=51$). The data source of this paper is CDC WONDER (https://wonder.cdc.gov/mcd.html (accessed on 21 April 2023)). An advantage of this dataset is its reliability as it was systematically recorded after the outbreak of the COVID-19 pandemic and it is publicly available.

We use the following variables: (i) the count of COVID-19 deaths; (ii) the count of causes of death that may be medically related to COVID-19; (iii) the count of the rest of the possible causes of death. Categories (ii) and (iii) include deaths that directly and indirectly, respectively, may be due to COVID-19. Nevertheless, the same categories also include deaths that are independent of COVID-19. We reviewed the underlying possible cause-of-death data (i.e., death codes) of the National Center for Health Statistics (NCHS) in detail, and we classified the different causes of death which we present in the Supplementary Materials. In

Table 1. Descriptive statistics for COVID-19 death count from March 2020 to September 2022.

	Minimum	Maximum	Mean	Standard Deviation	Skewness	Excess Kurtosis
Alabama	0	4584	$1045.28$	$1156.36$	$1.5133$	$1.5310$
Alaska	0	408	$67.11$	$95.57$	$2.1642$	$4.3524$
Arizona	0	7516	$1451.92$	$1729.42$	$1.7666$	$2.9192$
Arkansas	0	2122	$580.94$	$615.10$	$1.1415$	$0.1638$
California	0	37742	$5156.78$	$7367.66$	$3.0502$	$9.8742$
Colorado	0	2864	$675.11$	$729.10$	$1.4096$	$0.9624$
Connecticut	0	4720	$597.11$	$961.24$	$2.8141$	$8.2073$
Delaware	0	588	$149.72$	$169.85$	$1.4698$	$1.0488$
District of Columbia	0	524	$96.61$	$127.45$	$2.1445$	$4.2285$
Florida	0	19400	$3712.81$	$4233.49$	$2.1260$	$4.5780$
Georgia	0	7378	$1725.72$	$1882.12$	$1.6015$	$1.9995$
Hawaii	0	468	$79.33$	$99.77$	$2.1843$	$4.9951$
Idaho	0	1202	$257.39$	$307.64$	$1.7488$	$2.5641$
Illinois	0	7448	$1820.31$	$2019.78$	$1.5248$	$1.2143$
Indiana	0	5456	$1249.61$	$1348.98$	$1.4713$	$1.3869$
Iowa	0	2788	$502.06$	$608.78$	$2.2252$	$5.1321$
Kansas	0	2460	$476.44$	$578.61$	$1.8028$	$2.7065$
Kentucky	0	3348	$883.44$	$900.58$	$1.2612$	$0.4046$
Louisiana	0	3360	$844.72$	$874.59$	$1.3634$	$1.1153$
Maine	0	622	$137.78$	$155.84$	$1.5724$	$1.5953$
Maryland	0	3026	$794.28$	$859.28$	$1.4429$	$0.9031$
Massachusetts	0	7613	$978.83$	$1598.92$	$2.8906$	$8.1912$
Michigan	0	6939	$1689.47$	$1889.87$	$1.4313$	$0.9831$
Minnesota	0	3040	$649.11$	$729.85$	$1.7811$	$2.6115$
Mississippi	0	2314	$661.06$	$676.46$	$1.1450$	$0.1584$
Missouri	0	4301	$1112.11$	$1159.65$	$1.3452$	$0.8013$
Montana	0	828	$182.06$	$229.64$	$1.4750$	$0.9767$
Nebraska	0	1428	$250.89$	$310.94$	$2.1705$	$4.8834$
Nevada	0	2328	$601.56$	$608.92$	$1.3169$	$1.0377$
New Hampshire	0	542	$128.78$	$151.40$	$1.5648$	$1.3628$
New Jersey	0	17245	$1674.50$	$3059.54$	$3.9339$	$16.9425$
New Mexico	0	1964	$423.78$	$497.24$	$1.5762$	$1.5237$
New York	0	42119	$3777.97$	$7244.47$	$4.3289$	$20.1510$
North Carolina	0	6722	$1559.67$	$1604.09$	$1.5217$	$1.7279$
North Dakota	0	1014	$142.22$	$215.16$	$2.4802$	$6.3277$
Ohio	0	11064	$2282.22$	$2713.98$	$1.7141$	$2.1354$
Oklahoma	0	3582	$806.97$	$974.36$	$1.5646$	$1.4974$
Oregon	0	1700	$385.94$	$396.33$	$1.5650$	$1.7997$
Pennsylvania	0	11891	$2431.72$	$2829.49$	$1.6552$	$2.1690$
Rhode Island	0	856	$174.72$	$236.05$	$1.7743$	$1.9867$
South Carolina	0	3950	$965.06$	$1026.16$	$1.4107$	$1.3771$
South Dakota	0	1322	$161.39$	$255.67$	$3.1155$	$10.5952$
Tennessee	0	5794	$1423.81$	$1560.35$	$1.4597$	$1.1334$
Texas	0	20078	$5080.72$	$5439.40$	$1.1428$	$0.2174$
Utah	0	916	$269.00$	$267.87$	$1.0083$	$-0.4097$
Vermont	0	118	$32.28$	$34.62$	$1.0363$	$0.2831$
Virginia	0	4276	$1060.33$	$1018.11$	$1.4957$	$1.8726$
Washington	0	2216	$642.94$	$549.12$	$1.2333$	$0.6577$
West Virginia	0	1424	$372.33$	$419.76$	$1.2877$	$0.2590$
Wisconsin	0	3692	$745.28$	$916.81$	$1.7316$	$2.1409$
Wyoming	0	382	$81.56$	$110.23$	$1.4071$	$0.7375$

,

Table 2. Descriptive statistics for deaths medically related to COVID-19 count from March 2020 to September 2022.

	Minimum	Maximum	Mean	Standard Deviation	Skewness	Excess Kurtosis
Alabama	6813	13365	$8359.08$	$1576.13$	$1.5057$	$1.6061$
Alaska	156	682	$300.19$	$122.32$	$1.7061$	$2.5773$
Arizona	7339	16971	$9329.69$	$2254.07$	$1.7536$	$2.6881$
Arkansas	3727	6590	$4623.00$	$820.07$	$0.9809$	$0.0068$
California	35190	86199	$43356.94$	$10126.33$	$2.8427$	$8.4630$
Colorado	3889	7262	$4848.22$	$933.69$	$1.2054$	$0.2776$
Connecticut	2870	8766	$3882.69$	$1183.31$	$2.4397$	$6.5054$
Delaware	673	1746	$978.14$	$238.95$	$1.5206$	$2.0926$
District of Columbia	350	1073	$505.03$	$172.04$	$1.8322$	$2.5817$
Florida	29435	51950	$33682.86$	$5182.74$	$1.8696$	$3.0734$
Georgia	10524	21311	$13234.78$	$2561.23$	$1.5986$	$2.0099$
Hawaii	698	1723	$1068.03$	$191.64$	$1.3706$	$3.0313$
Idaho	1068	2621	$1469.75$	$362.75$	$1.5466$	$2.1371$
Illinois	13268	24308	$16134.61$	$2865.69$	$1.5159$	$1.3041$
Indiana	7680	14732	$9681.28$	$1768.70$	$1.3631$	$1.1049$
Iowa	3316	6311	$4138.19$	$787.13$	$1.4466$	$1.1282$
Kansas	2593	5775	$3375.83$	$764.03$	$1.6228$	$2.1014$
Kentucky	5676	9639	$6940.89$	$1149.51$	$1.0547$	$-0.1203$
Louisiana	5388	9992	$6877.78$	$1183.68$	$1.1874$	$0.5398$
Maine	1035	2323	$1504.72$	$292.32$	$1.1062$	$1.0899$
Maryland	5550	10608	$7025.56$	$1260.12$	$1.3814$	$1.0502$
Massachusetts	5879	15198	$7570.39$	$1890.65$	$2.4362$	$6.1814$
Michigan	13328	23583	$15943.64$	$2766.05$	$1.4310$	$1.0778$
Minnesota	4405	7996	$5421.92$	$977.71$	$1.4578$	$1.0278$
Mississippi	3817	7429	$4824.11$	$912.66$	$1.3112$	$0.9047$
Missouri	8090	14043	$9710.89$	$1582.19$	$1.3898$	$1.1299$
Montana	763	1917	$1103.56$	$307.77$	$1.0371$	$-0.0117$
Nebraska	1209	2934	$1716.72$	$399.19$	$1.3680$	$1.4514$
Nevada	3078	6406	$3988.31$	$815.12$	$1.3760$	$1.4885$
New Hampshire	959	1821	$1197.06$	$206.17$	$1.4804$	$1.5145$
New Jersey	8912	32665	$11528.58$	$4243.95$	$3.7002$	$15.3563$
New Mexico	1668	4169	$2318.11$	$634.68$	$1.4924$	$1.3102$
New York	20603	88497	$27266.36$	$11474.37$	$4.4111$	$20.9013$
North Carolina	11511	20744	$13861.31$	$2143.19$	$1.5848$	$2.0655$
North Dakota	406	1495	$603.00$	$232.01$	$2.1749$	$4.8300$
Ohio	15732	30833	$19122.81$	$3852.55$	$1.6975$	$2.0651$
Oklahoma	4846	9631	$6053.06$	$1309.18$	$1.4498$	$1.2202$
Oregon	3676	5864	$4375.42$	$638.75$	$1.0376$	$-0.1032$
Pennsylvania	16728	33050	$20365.47$	$4062.30$	$1.6309$	$1.8973$
Rhode Island	748	1811	$1069.75$	$289.56$	$1.3392$	$0.6765$
South Carolina	5704	11172	$7070.75$	$1304.47$	$1.4467$	$1.6760$
South Dakota	472	1946	$718.81$	$278.15$	$2.7081$	$8.8157$
Tennessee	9320	16491	$11541.25$	$2080.06$	$1.2942$	$0.6464$
Texas	27085	52118	$33668.28$	$6578.97$	$1.1282$	$0.2704$
Utah	1748	3032	$2104.67$	$356.27$	$1.0996$	$-0.0306$
Vermont	394	742	$518.61$	$86.69$	$0.5832$	$-0.2076$
Virginia	8315	15027	$10038.36$	$1451.20$	$1.7607$	$3.2330$
Washington	6005	10031	$7394.22$	$890.70$	$1.0818$	$0.8547$
West Virginia	2218	4285	$3014.03$	$607.08$	$0.9663$	$-0.4568$
Wisconsin	6010	11132	$7332.89$	$1310.12$	$1.5493$	$1.4493$
Wyoming	214	718	$382.86$	$138.20$	$1.1118$	$0.3274$

and

Table 3. Descriptive statistics for the rest of the deaths count from March 2020 to September 2022.

	Minimum	Maximum	Mean	Standard Deviation	Skewness	Excess Kurtosis
Alabama	1342	2355	$2001.75$	$222.33$	$-1.4161$	$2.2990$
Alaska	10	102	$44.42$	$26.84$	$0.4216$	$-1.0762$
Arizona	1929	4082	$3218.19$	$464.02$	$-1.0833$	$1.5031$
Arkansas	759	1188	$1023.72$	$109.08$	$-0.6110$	$0.0486$
California	9494	16663	$14268.53$	$1586.18$	$-1.6058$	$2.9274$
Colorado	1212	2439	$2006.36$	$273.26$	$-1.3818$	$1.8621$
Connecticut	869	1447	$1275.44$	$131.35$	$-1.4145$	$2.0187$
Delaware	133	313	$245.53$	$43.84$	$-0.7893$	$0.0369$
District of Columbia	36	252	$166.36$	$53.33$	$-0.7107$	$0.6959$
Florida	6331	11686	$10540.58$	$1291.85$	$-2.3599$	$4.8058$
Georgia	2670	4861	$3970.67$	$464.64$	$-1.4130$	$2.5393$
Hawaii	195	376	$266.50$	$44.28$	$0.4743$	$-0.1742$
Idaho	292	548	$402.19$	$65.90$	$0.3429$	$-0.8229$
Illinois	3297	5617	$4776.94$	$512.50$	$-1.5449$	$2.7427$
Indiana	1801	3257	$2792.83$	$318.82$	$-1.5776$	$2.6958$
Iowa	658	1105	$912.25$	$99.13$	$-0.6511$	$0.3132$
Kansas	577	1195	$918.44$	$124.85$	$-0.4404$	$0.5202$
Kentucky	1275	2474	$2057.83$	$281.51$	$-1.3303$	$1.7042$
Louisiana	1182	2478	$2058.19$	$304.20$	$-1.6985$	$2.5003$
Maine	315	633	$503.36$	$71.51$	$-0.5161$	$0.3990$
Maryland	1319	2519	$2085.19$	$267.69$	$-1.5961$	$2.8379$
Massachusetts	1671	3086	$2607.39$	$288.49$	$-1.3104$	$2.7209$
Michigan	2648	4611	$4073.19$	$457.21$	$-1.8441$	$3.2215$
Minnesota	1317	2259	$1891.08$	$209.04$	$-0.9916$	$1.2693$
Mississippi	680	1263	$1060.56$	$140.13$	$-1.2451$	$1.3818$
Missouri	1787	3297	$2781.94$	$319.62$	$-1.7232$	$3.4969$
Montana	141	367	$222.61$	$52.62$	$0.6212$	$0.1786$
Nebraska	274	546	$426.28$	$75.36$	$-0.0935$	$-0.9619$
Nevada	636	1416	$997.53$	$162.36$	$0.1399$	$0.4362$
New Hampshire	237	450	$342.44$	$54.66$	$-0.2250$	$-0.4060$
New Jersey	2190	4084	$3133.75$	$357.07$	$-0.2976$	$1.8624$
New Mexico	437	1090	$812.03$	$153.74$	$-0.7699$	$0.0826$
New York	4451	7516	$6480.19$	$682.60$	$-1.6806$	$3.0609$
North Carolina	2837	5231	$4527.00$	$535.25$	$-2.0827$	$4.0843$
North Dakota	26	144	$74.89$	$26.06$	$0.5768$	$0.2763$
Ohio	3457	6415	$5550.28$	$668.81$	$-1.8173$	$3.2664$
Oklahoma	946	1914	$1510.78$	$206.37$	$-1.2191$	$1.7029$
Oregon	1122	1890	$1529.33$	$184.66$	$-0.4451$	$-0.4135$
Pennsylvania	4001	7087	$6120.78$	$693.22$	$-1.9731$	$3.7873$
Rhode Island	118	414	$298.42$	$59.47$	$-0.8882$	$1.3203$
South Carolina	1428	2864	$2358.97$	$298.75$	$-1.6302$	$2.4604$
South Dakota	30	211	$117.47$	$40.83$	$0.0011$	$-0.4474$
Tennessee	1962	3979	$3499.47$	$508.18$	$-2.0449$	$3.4816$
Texas	7085	12089	$10368.08$	$1096.24$	$-1.6808$	$2.7130$
Utah	449	896	$727.81$	$107.78$	$-0.8264$	$0.0558$
Vermont	32	174	$97.17$	$36.03$	$0.2359$	$-0.7916$
Virginia	2115	3787	$3169.31$	$358.54$	$-1.5807$	$2.6412$
Washington	1799	3204	$2669.25$	$333.01$	$-0.7986$	$0.4624$
West Virginia	352	1073	$823.67$	$151.68$	$-1.1885$	$1.6153$
Wisconsin	1536	2825	$2311.11$	$277.86$	$-1.2120$	$1.7742$
Wyoming	14	78	$36.53$	$15.94$	$0.8644$	$0.1107$

, we present descriptive statistics for each US state for (i), (ii), and (iii), respectively. These show the minimum, maximum, mean, sample standard deviation (SD), skewness, and excess kurtosis of death counts for each state from March 2020 to September 2022. The time-varying unobserved effects component ${\mu }_{i,t}$ of the panel data model captures the variation among US states, shown in Table 1, Table 2 and Table 3, through the US state-specific score function vector $u_{i,t}$.

In Figure 1, we present the evolution of US death counts for categories (i)–(iii), i.e., ${\sum }_{i=1}^N{y}_{1,i,t}$ for COVID-19 deaths, ${\sum }_{i=1}^N{y}_{2,i,t}$ for causes of death that may be medically related to COVID-19, and ${\sum }_{i=1}^N{y}_{3,i,t}$ for the rest of the causes of death, from January 2018 to September 2022. In Figure 1, we indicate with a vertical dashed line the start date of the COVID-19 pandemic, i.e., March 2020.

We performed a heteroscedasticity and autocorrelation consistent (HAC) t-test (Newey and West 1987) for ${\sum }_{i=1}^N{y}_{2,i,t}$ and ${\sum }_{i=1}^N{y}_{3,i,t}$ to compare the mean total US death counts from before the pandemic. The corresponding test statistics are ${11.1836}^{\ast \ast \ast }$ for the causes of death that may be medically related to COVID-19 and ${3.8155}^{\ast \ast \ast }$ for the rest of the causes of death. This indicates that due to the COVID-19 pandemic, deaths in the categories that include causes of death medically related to COVID-19 and causes of death medically unrelated to COVID-19 simultaneously increased. We also highlight the peaks in April 2020, July 2020, January 2021, September 2021, and January 2022, which can be simultaneously observed in all panels of Figure 1, indicating significant positive association among the elements of $y_{i,t}$. Figure 1 indicates peaks and non-stationary behavior for categories (i)–(iii). Therefore, the diagnostic tests for t-PQVAR are important to justify the model specification. We show that the ML consistency and asymptotic normality conditions are supported.

3.2. Methods

3.2.1. t-PQVAR(p)

PQVAR extends the classical PVAR model with time-invariant unobserved effects $y_{i,t}=c+{\mu }_i+v_{i,t}$ (i.e., panel error-component model) as we consider score-driven time variation in the unobserved effects component and use $y_{i,t}=c+{\mu }_{i,t}+v_{i,t}$. The score-driven filter ${\mu }_{i,t}$ captures all exogenous individual-specific explanatory variables. All terms in PQVAR are column vectors with elements corresponding to the vector of dependent variable $y_{i,t}$, and we assume that $v_{i,t}$ has a multivariate t distribution. The score-driven updating mechanism of unobserved effects ${\mu }_{i,t}$ improves the model specification and the efficiency of parameter estimates because it implies an information-theoretically optimal multivariate filter (Blasques et al. 2015) and robustness to extreme observations (Harvey 2013). This is relevant because we use a small sample, and a better dynamic specification can provide more reliable estimates.

The t-PQVAR(p) model for death counts $y_{i,t}$$(K\times 1)$ is given by

$$y_{i,t}=c+{\mu }_{i,t}+v_{i,t}$$

(1)

$${\mu }_{i,t}={\Phi }_1{\mu }_{i,t-1}+\dots +{\Phi }_p{\mu }_{i,t-p}+{\Psi }_1{u}_{i,t-1}$$

(2)

for US states $i=1,\dots ,N$ and periods $t=1,\dots ,T$, where c$(K\times 1)$, ${\Phi }_1,\dots ,{\Phi }_p$ (each $K\times K$), and ${\Psi }_1$$(K\times K)$ are time-invariant parameters, $v_{i,t}$ ($K\times 1$) is the reduced-form error term, and $u_{i,t}$$(K\times 1)$ is a scaled score function vector. Dynamic unobserved heterogeneity is captured by ${\mu }_{i,t}$. For the first p observations, we initialize ${\mu }_{i,t}$ by using ${\mu }_{i,t}=E\left({\mu }_{i,t}\right)=0_{K\times 1}$. Variable $v_{i,t}$ is an i.i.d. reduced-form error term with $v_{i,t}\sim t_K(0,\Sigma ,\nu )$, where the scale matrix $\Sigma ={\Omega }^{-1}{\left({\Omega }^{-1}\right)}^{\prime }$ is positive definite, ${\Omega }^{-1}{\left({\Omega }^{-1}\right)}^{\prime }$ is the unique Cholesky decomposition of $\Sigma$, ${\Omega }^{-1}$$(K\times K)$ is a lower triangular matrix, and the degrees of freedom parameter is $\nu >2$ (hence, the covariance of $v_{i,t}$ exists). In Equation (1), we do not use exogenous explanatory variables because our objective is the measurement of the dynamic interaction effects among the dependent variables within $y_{i,t}$. Hence, several conditions for the PVAR models (Alvarez and Arellano 2003; Holtz-Eakin et al. 1988) simplify for PQVAR.

The log conditional density of $y_{i,t}$ is

$$lnf\left(y_{i,t}\right|y_{i,1},\dots ,y_{i,t-1})=ln\Gamma \left({\displaystyle \frac{\nu +K}{2}}\right)-ln\Gamma \left({\displaystyle \frac{\nu }{2}}\right)-{\displaystyle \frac{K}{2}}ln\left(\pi \nu \right)$$

(3)

$$-{\displaystyle \frac{1}{2}}ln|\Sigma |-{\displaystyle \frac{\nu +K}{2}}ln\left(1+{\displaystyle \frac{v_{i,t}^{\prime }{\Sigma }^{-1}{v}_{i,t}}{\nu }}\right).$$

The partial derivative of the log of the conditional density for ${\mu }_{i,t}$ is

$${\displaystyle \frac{\mathsf{\partial }lnf\left(y_{i,t}\right|y_{i,1},\dots ,y_{i,t-1})}{\mathsf{\partial }{\mu }_{i,t}}}={\displaystyle \frac{\nu +K}{\nu }}{\Sigma }^{-1}\times {\left(1+{\displaystyle \frac{v_{i,t}^{\prime }{\Sigma }^{-1}{v}_{i,t}}{\nu }}\right)}^{-1}{v}_{i,t}={\displaystyle \frac{\nu +K}{\nu }}{\Sigma }^{-1}\times u_{i,t}.$$

(4)

The last equal sign in Equation (4) defines the scaled score function $u_{i,t}$ (Harvey 2013) using $v_{i,t}$, where $v_{i,t}$ is multiplied by the variable ${[1+\left(v_{i,t}^{\prime }{\Sigma }^{-1}{v}_{i,t}\right)/\nu ]}^{-1}=\nu /(\nu +v_{i,t}^{\prime }{\Sigma }^{-1}{v}_{i,t})\in (0,1)$. Therefore, the scaled score function is bounded by $v_{i,t}$. Hence, the conditional mean updates of PQVAR are more robust to extreme observations than PVARMA or PVAR.

The scaled score function $u_{i,t}$ is multivariate i.i.d. with a mean zero and covariance matrix:

$$\mathrm{Var}\left(u_{i,t}\right)=E\left[{\displaystyle \frac{\mathsf{\partial }lnf\left(y_{i,t}\right|y_{i,1},\dots ,y_{i,t-1})}{\mathsf{\partial }{\mu }_{i,t}}}\times {\displaystyle \frac{\mathsf{\partial }lnf\left(y_{i,t}\right|y_{i,1},\dots ,y_{i,t-1})}{\mathsf{\partial }{\mu }_{i,t}^{\prime }}}\right]={\displaystyle \frac{\nu +K}{\nu +K+2}}{\Sigma }^{-1}.$$

(5)

The unconditional mean and variance of $v_{i,t}$ are $E\left(v_{i,t}\right)=0$ and $\mathrm{Var}\left(v_{i,t}\right)=\Sigma \times \nu /(\nu -2)$, respectively.

We factorize the variance $\mathrm{Var}\left(v_{i,t}\right)$ as

$$\mathrm{Var}\left(v_{i,t}\right)={\displaystyle \frac{\nu }{\nu -2}}\times \Sigma ={\left({\displaystyle \frac{\nu }{\nu -2}}\right)}^{1/2}\times {\Omega }^{-1}{\left({\Omega }^{-1}\right)}^{\prime }\times {\left({\displaystyle \frac{\nu }{\nu -2}}\right)}^{1/2},$$

(6)

and we introduce the multivariate i.i.d. structural-form error term ${ϵ}_{i,t}$ as

$$v_{i,t}={\left({\displaystyle \frac{\nu }{\nu -2}}\right)}^{1/2}{\Omega }^{-1}\times {ϵ}_{i,t},$$

(7)

where $E\left({ϵ}_{i,t}\right)=0$, $\mathrm{Var}\left({ϵ}_{i,t}\right)=I_K$, and ${ϵ}_{i,t}\sim t_K[0,I_K\times (\nu -2)/\nu ,\nu ]$. We substitute Equation (7) into the scaled score function $u_{i,t}$ of Equation (4) to obtain the structural-form error term representation of $u_{i,t}$:

$$u_{i,t}={\left[(\nu -2)\nu \right]}^{1/2}{\Omega }^{-1}\times {\displaystyle \frac{{ϵ}_{i,t}}{\nu -2+{ϵ}_{i,t}^{\prime }{ϵ}_{i,t}}}$$

(8)

which shows that $u_{i,t}$ is a bounded function of ${ϵ}_{i,t}$, hence all moments of $u_{i,t}$ exist. We use the results on the structural-form error representations of $v_{i,t}$ and $u_{i,t}$ to formulate the IRFs in Appendix A.

In the results section, we report the p-values of the Escanciano–Lobato test (Escanciano and Lobato 2009) and the Ljung–Box test (Ljung and Box 1978) of ${ϵ}_{i,t}$ among model diagnostics. The null hypothesis of the Escanciano–Lobato test is that the time series variable is a martingale difference sequence (MDS), and the opposite is the alternative hypothesis. The null hypothesis of the Ljung–Box test is an independent sequence of random variables, and the opposite is the alternative hypothesis.

3.2.2. Estimation of the t-PQVAR(p) Model

The t-PQVAR model is estimated using the ML method:

$${\widehat{\Theta }}_{\mathrm{ML}}=arg\underset{\Theta }{max}\mathrm{LL}(y_{i,t}:i=1,\dots ,N,t=1,\dots ,T;\Theta )$$

(9)

$$=arg\underset{\Theta }{max}\sum _{i=1}^N\sum _{t=1}^{T}lnf\left(y_{i,t}\right|y_{i,1},\dots ,y_{i,t-1};\Theta ),$$

where $\Theta$ are the parameters and LL is the log-likelihood (Blasques et al. 2022; Creal et al. 2013; Harvey 2013). We maximize the LL numerically and use the inverse information matrix to estimate standard errors. The asymptotic theory assumes that N is fixed and $T\to \infty$. We present the ML conditions of consistency and asymptotic normality in Appendix B.

3.2.3. The Gaussian-PVARMA(p,p) Model

Classical linear Gaussian time series models are special cases of the t-PQVAR model. If $\nu \to \infty$, then $v_{i,t}\sim t_K(0,\Sigma ,\nu ){\to }_d{N}_K(0,\Sigma )$ and $u_{i,t}=v_{i,t}{[1+\left(v_{i,t}^{\prime }{\Sigma }^{-1}{v}_{i,t}\right)/\nu ]}^{-1}{\to }_p{v}_{i,t}$. Hence,

$$y_{i,t}=c-{\Phi }_{1}c-\dots -{\Phi }_{p}c+{\Phi }_1{y}_{i,t-1}+\dots +{\Phi }_p{y}_{i,t-p}+v_{i,t}+{\Psi }_1{v}_{i,t-1}-{\Phi }_1{v}_{i,t-1}-\dots -{\Phi }_p{v}_{i,t-p},$$

(10)

which is a Gaussian-PVARMA(p,p) model. For the limiting case of t-PQVAR(1), we have

$$y_{i,t}=c-{\Phi }_{1}c+{\Phi }_1{y}_{i,t-1}+v_{i,t}+({\Psi }_1-{\Phi }_1)v_{i,t-1},$$

(11)

which is a Gaussian-PVARMA(1,1) model. For ${\Psi }_1={\Phi }_1$, we have the Gaussian-PVAR(1) model:

$$y_{i,t}=c-{\Phi }_{1}c+{\Phi }_1{y}_{i,t-1}+v_{i,t}.$$

(12)

The Gaussian-PVARMA(p,p) and Gaussian-PVAR(p) specifications are special cases of Gaussian-PVARMA(p,q) (Lütkepohl 2005), frequently used in practice to measure dynamic interactions for time series data. For the Gaussian-PVARMA(p,q) model, Lütkepohl (2005) presents the computation of the IRF and the ML estimation with the conditions of the consistency and asymptotic normality of ML. Using $C_1$ and $C_2$, we denote the statistics for the covariance stationarity and invertibility, respectively, of the Gaussian-PVARMA(p,q) model (Lütkepohl 2005). Covariance stationarity and invertibility are supported if $C_1<1$ and $C_2<1$, respectively.

3.2.4. Model Specification and IRF Identification

We use the multivariate t distribution, instead of a discrete distribution, to measure dynamic interactions among the death counts of different possible causes. One alternative could be the multivariate Poisson distribution (Marshall and Olkin 1985). However, the multivariate Poisson distribution implies more restrictions on interactions than the multivariate t distribution (Geenens 2020). Another alternative could be copulas to associate discrete (e.g., Poisson or negative binomial) marginal distributions. Geenens (2020) suggests the iterated proportional fitting (IPF) procedure to associate discrete random variables using continuous copulas. We studied this procedure for our case, but found that it cannot be applied in a straightforward way to score-driven models. Concerning the approximation using the continuous multivariate t distribution, if a Poisson distribution has a sufficiently large expected value, then it can be approximated well by a normal distribution. As the normal distribution is a special case of the Student’s t, our approximation may work. To check its robustness, we compare the statistical performances of the multivariate normal and t distributions.

We consider alternative lag structures for the t-PQVAR, Gaussian-PVARMA, and Gaussian-PVAR models. We report results for the t-PQVAR(1), Gaussian-PVARMA(1,1), and Gaussian-PVAR(1) specifications that are supported by the ML conditions and diagnostic test results. For (i) COVID-19 mortality $y_{1,i,t}$, (ii) causes of deaths that may be medically related to COVID-19 $y_{2,i,t}$, and (iii) the rest of the causes $y_{3,i,t}$, we estimate the following t-PQVAR(1) model:

$$\left[\begin{array}{c}{y}_{1,i,t}\\ y_{2,i,t}\\ y_{3,i,t}\end{array}\right]=\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\end{array}\right]+\left[\begin{array}{ccc}{\Phi }_{1,11}& {\Phi }_{1,12}& {\Phi }_{1,13}\\ {\Phi }_{1,21}& {\Phi }_{1,22}& {\Phi }_{1,23}\\ {\Phi }_{1,31}& {\Phi }_{1,32}& {\Phi }_{1,33}\end{array}\right]\left[\begin{array}{c}{\mu }_{1,i,t-1}\\ {\mu }_{2,i,t-1}\\ {\mu }_{3,i,t-1}\end{array}\right]+$$

(13)

$$\left[\begin{array}{ccc}{\Psi }_{1,11}& {\Psi }_{1,12}& {\Psi }_{1,13}\\ {\Psi }_{1,21}& {\Psi }_{1,22}& {\Psi }_{1,23}\\ {\Psi }_{1,31}& {\Psi }_{1,32}& {\Psi }_{1,33}\end{array}\right]\left[\begin{array}{c}{u}_{1,i,t-1}\\ u_{2,i,t-1}\\ u_{3,i,t-1}\end{array}\right]+\left[\begin{array}{c}{v}_{1,i,t}\\ v_{2,i,t}\\ v_{3,i,t}\end{array}\right]$$

The covariance matrix of the error term is $\mathrm{Var}\left(v_{i,t}\right)=\Sigma \times [\nu /(\nu -2)]={\Omega }^{-1}{\left({\Omega }^{-1}\right)}^{\prime }\times [\nu /(\nu -2)]$, where

$${\Omega }^{-1}=\left[\begin{array}{ccc}{\Omega }_{11}^{-1}& 0& 0\\ {\Omega }_{21}^{-1}& {\Omega }_{22}^{-1}& 0\\ {\Omega }_{31}^{-1}& {\Omega }_{32}^{-1}& {\Omega }_{33}^{-1}\end{array}\right]$$

(14)

and we use the Cholesky decomposition $\Sigma ={\Omega }^{-1}{\left({\Omega }^{-1}\right)}^{\prime }$. Then,

$$\left[\begin{array}{c}{v}_{1,i,t}\\ v_{2,i,t}\\ v_{3,i,t}\end{array}\right]={\left({\displaystyle \frac{\nu }{\nu -2}}\right)}^{1/2}{\Omega }^{-1}\left[\begin{array}{c}{ϵ}_{1,i,t}\\ {ϵ}_{2,i,t}\\ {ϵ}_{3,i,t}\end{array}\right]={\left({\displaystyle \frac{\nu }{\nu -2}}\right)}^{1/2}\left[\begin{array}{c}{\Omega }_{11}^{-1}{ϵ}_{1,i,t}\hfill \\ {\Omega }_{21}^{-1}{ϵ}_{1,i,t}+{\Omega }_{22}^{-1}{ϵ}_{2,i,t}\hfill \\ {\Omega }_{31}^{-1}{ϵ}_{1,i,t}+{\Omega }_{32}^{-1}{ϵ}_{2,i,t}+{\Omega }_{33}^{-1}{ϵ}_{3,i,t}\hfill \end{array}\right]$$

(15)

We estimate the IRFs using 10,000 Monte Carlo simulations of $\Omega$ under sign restrictions on contemporaneous effects (Rubio-Ramírez et al. 2010). First, we use the ML estimates of $\Omega$. Second, we simulate an $K\times K$ matrix $\tilde{K}$ of i.i.d. $N(0,1)$ numbers. Third, we compute the QR decomposition of $\tilde{K}$ and denote the resulting matrices $\tilde{Q}$ and $\tilde{R}$. Fourth, we define $\tilde{\Omega }\equiv \Omega \times {\tilde{Q}}^{\prime }$ for each simulation. We replace the matrix $\Omega$ with $\tilde{\Omega }$ in the IRFs for each simulation.

Concerning the sign restrictions for contemporaneous effects among $(v_{1,i,t},v_{2,i,t},v_{3,i,t})$, for each simulation of $\tilde{\Omega }$, we assume positive sign restrictions for all contemporaneous relationships among causes of death in categories (i)–(iii). These restrictions are motivated by the literature on COVID-19 and non-COVID-19 mortality (see the papers cited in Section 2.1). The advantage of the sign restrictions-based IRF identification, compared to the recursive IRF identification (Kilian and Lütkepohl 2017) that uses the Cholesky decomposition of the covariance matrix of $v_{i,t}$, is that we do not restrict any contemporaneous interaction effects among the reduced-form error terms $v_{i,t}$ to zero.

4. Results

We present ML diagnostics and parameter estimates for the t-PQVAR(1), Gaussian-PVARMA(1,1), and Gaussian-PVAR(1) models in

Table 4. Parameter estimates and model diagnostics.

	t-PQVAR(1)	Gaussian-PVARMA(1,1)	Gaussian-PVAR(1)
${\Phi }_{1,11}$	${0.5153}^{\ast \ast \ast }\left(0.0108\right)$	${0.6343}^{\ast \ast \ast }\left(0.0079\right)$	${0.6176}^{\ast \ast \ast }\left(0.0047\right)$
${\Phi }_{1,12}$	${0.4049}^{\ast \ast }\left(0.1801\right)$	$0.0802\left(0.2355\right)$	${0.4834}^{\ast \ast \ast }\left(0.0694\right)$
${\Phi }_{1,13}$	$0.0459\left(0.1497\right)$	$0.2506\left(0.2040\right)$	$-0.0635\left(0.0648\right)$
${\Phi }_{1,21}$	$-{0.0098}^{\ast \ast \ast }\left(0.0031\right)$	$-{0.0065}^{\ast }\left(0.0036\right)$	$-{0.0102}^{\ast \ast \ast }\left(0.0023\right)$
${\Phi }_{1,22}$	${0.8664}^{\ast \ast \ast }\left(0.0346\right)$	${0.8230}^{\ast \ast \ast }\left(0.0338\right)$	${0.9038}^{\ast \ast \ast }\left(0.0115\right)$
${\Phi }_{1,23}$	${0.1202}^{\ast \ast \ast }\left(0.0292\right)$	${0.1556}^{\ast \ast \ast }\left(0.0266\right)$	${0.0880}^{\ast \ast \ast }\left(0.0084\right)$
${\Phi }_{1,31}$	$-0.0014\left(0.0023\right)$	$-0.0002\left(0.0020\right)$	$-{0.0089}^{\ast \ast }\left(0.0035\right)$
${\Phi }_{1,32}$	${0.0949}^{\ast \ast \ast }\left(0.0168\right)$	${0.0661}^{\ast \ast \ast }\left(0.0177\right)$	${0.2540}^{\ast \ast \ast }\left(0.0170\right)$
${\Phi }_{1,33}$	${0.9155}^{\ast \ast \ast }\left(0.0137\right)$	${0.9421}^{\ast \ast \ast }\left(0.0139\right)$	${0.7772}^{\ast \ast \ast }\left(0.0102\right)$
${\Omega }_{1,1}^{-1}$	${0.6544}^{\ast \ast \ast }\left(0.0131\right)$	${0.9994}^{\ast \ast \ast }\left(0.0163\right)$	${1.0384}^{\ast \ast \ast }\left(0.0088\right)$
${\Omega }_{2,1}^{-1}$	${0.0860}^{\ast \ast \ast }\left(0.0066\right)$	${0.0845}^{\ast \ast \ast }\left(0.0043\right)$	${0.0835}^{\ast \ast \ast }\left(0.0024\right)$
${\Omega }_{2,2}^{-1}$	${0.1074}^{\ast \ast \ast }\left(0.0030\right)$	${0.1406}^{\ast \ast \ast }\left(0.0022\right)$	${0.1459}^{\ast \ast \ast }\left(0.0010\right)$
${\Omega }_{3,1}^{-1}$	${0.0394}^{\ast \ast \ast }\left(0.0041\right)$	${0.0383}^{\ast \ast \ast }\left(0.0120\right)$	${0.0247}^{\ast \ast \ast }\left(0.0057\right)$
${\Omega }_{3,2}^{-1}$	${0.0180}^{\ast \ast \ast }\left(0.0057\right)$	${0.0307}^{\ast \ast \ast }\left(0.0061\right)$	${0.0561}^{\ast \ast \ast }\left(0.0055\right)$
${\Omega }_{3,3}^{-1}$	${0.1304}^{\ast \ast \ast }\left(0.0023\right)$	${0.2024}^{\ast \ast \ast }\left(0.0028\right)$	${0.2262}^{\ast \ast \ast }\left(0.0018\right)$
$c_1$	$-{0.8697}^{\ast }\left(0.4498\right)$	$-0.2997\left(0.7077\right)$	$-{1.3548}^{\ast \ast \ast }\left(0.2071\right)$
$c_2$	${0.3195}^{\ast \ast \ast }\left(0.0765\right)$	${0.4168}^{\ast \ast \ast }\left(0.1060\right)$	${0.2426}^{\ast \ast \ast }\left(0.0371\right)$
$c_3$	$-{0.1992}^{\ast \ast \ast }\left(0.0393\right)$	$-{0.1502}^{\ast \ast \ast }\left(0.0492\right)$	$-{0.5085}^{\ast \ast \ast }\left(0.0665\right)$
${\Psi }_{1,11}$	${1.5578}^{\ast \ast \ast }\left(0.0503\right)$	$-0.0133\left(0.0292\right)$	NA
${\Psi }_{1,12}$	${2.1889}^{\ast \ast \ast }\left(0.2866\right)$	${1.8799}^{\ast \ast \ast }\left(0.2629\right)$	NA
${\Psi }_{1,13}$	$0.0376\left(0.2730\right)$	$0.1805\left(0.3626\right)$	NA
${\Psi }_{1,21}$	${0.0226}^{\ast \ast }\left(0.0115\right)$	${0.0362}^{\ast \ast \ast }\left(0.0065\right)$	NA
${\Psi }_{1,22}$	${1.7163}^{\ast \ast \ast }\left(0.0767\right)$	$0.0073\left(0.0274\right)$	NA
${\Psi }_{1,23}$	$-{0.3642}^{\ast \ast \ast }\left(0.0650\right)$	$-{0.1661}^{\ast \ast \ast }\left(0.0304\right)$	NA
${\Psi }_{1,31}$	${0.0276}^{\ast \ast \ast }\left(0.0102\right)$	${0.0294}^{\ast \ast \ast }\left(0.0069\right)$	NA
${\Psi }_{1,32}$	$-{0.3962}^{\ast \ast \ast }\left(0.0487\right)$	$-{0.1661}^{\ast \ast \ast }\left(0.0335\right)$	NA
${\Psi }_{1,33}$	${1.1182}^{\ast \ast \ast }\left(0.0706\right)$	$-{0.6012}^{\ast \ast \ast }\left(0.0221\right)$	NA
$\nu$	${4.1695}^{\ast \ast \ast }\left(0.2329\right)$	NA	NA
LL	$-\mathbf{13.0161}$	$-35.5233$	$-45.0460$
AIC	$\mathbf{27.5877}$	$72.5466$	$91.0920$
BIC	$\mathbf{28.8193}$	$73.7342$	$91.8838$
HQC	$\mathbf{28.0175}$	$72.9611$	$91.3684$
$C_1$	$0.9963$	$0.9984$	$0.9924$
Mean $C_2$	NA	NA	NA
Mean $C_3$	$0.7593$	NA	NA
Mean $C_4$	$0.7285$	NA	NA
Mean $C_5$	$-0.9064$	$0.6833$	NA
Mean p-value, Escanciano–Lobato test ${ϵ}_{1,t}$	$0.4786$	$0.6892$	$0.4012$
Mean p-value, Escanciano–Lobato test ${ϵ}_{2,t}$	$0.3376$	$0.2246$	$0.3110$
Mean p-value, Escanciano–Lobato test ${ϵ}_{3,t}$	$0.3833$	$0.2478$	$0.2888$
Mean p-value, Ljung–Box test ${ϵ}_{1,t}$	$0.5140$	$0.6584$	$0.4740$
Mean p-value, Ljung–Box test ${ϵ}_{2,t}$	$0.4567$	$0.4159$	$0.4827$
Mean p-value, Ljung–Box test ${ϵ}_{3,t}$	$0.5627$	$0.3422$	$0.4265$

Panel quasi-vector autoregressive (PQVAR); panel VAR moving average (PVARMA); not available (NA); log-likelihood (LL); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan–Quinn criterion (HQC). Bold LL, AIC, BIC, and HQC model selection metrics indicate superior statistical performances. For all models, $C_1<1$ indicates covariance stationarity, i.e., the maximum modulus of the eigenvalues of ${\Phi }_1$ is lower than 1. For $C_2$ to $C_5$, `Mean’ indicates that for PQVAR, the average of each $C_2$ to $C_5$ is computed across all states of the United States (US). We do not report $C_2$ for PQVAR in this table, although the estimates support it. For PQVAR, Mean $C_3<1$ and Mean $C_4<1$ indicate that the maximum moduli of the eigenvalues of $E\left(X_{i,t}\right)$ and $E(X_{i,t}\otimes X_{i,t})$, respectively, are less than 1, which support the stability of the gradient and Hessian for the maximum likelihood (ML) estimator, on average, for the US states. For PQVAR, Mean $C_5<0$ indicates that invertibility is supported for the US. For PVARMA, $C_5<1$ indicates invertibility, i.e., the maximum modulus of the eigenvalues of ${\Psi }_1$ is lower than 1. Standard errors are reported in parentheses. $*$, $**$, and $***$ indicate significance at the 10%, 5%, and 1% levels, respectively.

. We find that $C_1$, $C_2$, $C_3$, $C_4$, and $C_5$ support the ML estimator for all models. For the structural-form errors, we report the mean p-values of the Escanciano–Lobato and Ljung–Box tests, which, under the null hypotheses, test whether the structural-form errors are martingale difference sequences (MDSs) and independent sequences, respectively. We computed the p-values for each state, and we report the mean p-values across states in Table 4. The Escanciano–Lobato and Ljung–Box test results support the lag-order specifications of the t-PQVAR, Gaussian-PVARMA, and Gaussian-PVAR models. Furthermore, we compare the statistical performance of the alternative models using the log-likelihood (LL) and Akaike information criterion (AIC), Bayesian information criterion (BIC), and Hannan–Quinn criterion (HQC) metrics (Harvey 2013) (see in Table 4), which suggest that the t-PQVAR(1) is superior to the Gaussian-PVARMA(1,1) and Gaussian-PVAR(1) models.

In Figure 2, Figure 3 and Figure 4, we present the IRFs for t-PQVAR(1), Gaussian-PVARMA(1,1), and Gaussian-PVAR(1), respectively. We present the interactions among (i) COVID-19 deaths (denoted as COVID-19 mortality); (ii) deaths that medically may be related to COVID-19 (denoted as COVID-19-med mortality); and (iii) the rest of the possible causes of death (denoted as non-COVID-19 mortality).

Figure 2, Figure 3 and Figure 4 show that the IRF of t-PQVAR(1) is more precise than the IRF of Gaussian-PVARMA(1,1) and Gaussian-PVAR(1). Therefore, we discuss the dynamic interaction effects for the t-PQVAR(1) model, instead of Gaussian-PVARMA(1,1) and Gaussian-PVAR(1), in the following section. These indicate significant positive dynamic effects that may be useful for forecasting purposes, for example when the PQVAR model of this paper is applied to the “Disease X” pandemic.

5. Discussion

In this section, we discuss the results of the IRF estimates and their generalizability to other countries or regions of the world for the t-PQVAR(1) model. First, we study contemporaneous correlations. We found that when 1000 persons die of COVID-19, about 100 persons die of possible causes that are medically related to COVID-19 in the same month (Figure 2, Panel D). This positive relationship may be due to lung, respiratory, heart, prostate, and diabetes diseases (e.g., J44.0, J44.1, J44.9, I27.9, J96.0, J96.1, and J96.9), which are caused by existing conditions severely influenced contemporaneously by a COVID-19 infection. The abbreviations for diseases in parentheses indicate the underlying cause of death codes of the NCHS (see their classification concerning COVID-19 in Supplementary Materials).

We also found that when 1000 persons die of COVID-19, about 100 persons die due to the rest of the possible causes (i.e., non-COVID-19 mortality) in the same month (Figure 2G). This relationship may be due to (i) isolation due to a fear of contracting COVID-19 or not wanting to burden the US medical services at the time of the pandemic, (ii) due to the pressure on healthcare services, patients with serious and potentially mortal diseases that are medically unrelated with COVID-19, such as Alzheimer’s or Parkinson’s disease, are unable to access medical services, (iii) due to the shutdown of the economy, deaths of despair (e.g., due to drug overdose, alcohol-related liver disease, and suicide), murders, or, for example, mortal accidents of unattended children during a period when the schools are closed. Specifically, the shutdown of the economy and isolation-focused protocols reduced exposure to sunlight, hence reducing vitamin D uptake, which potentially adversely affected the immune response to many non-COVID-19, non-medically related diseases (Tomaszewska et al. 2022). In addition to this, income losses from the economic shutdown increased financial strain, thus proliferating mental despair (Hertz-Palmor et al. 2021) and food insecurity (Yenerall and Jensen 2022).

On the other hand, when 1000 persons die of possible causes that are medically related to COVID-19, about 500 persons die of COVID-19 in the same month (Figure 2B). This may highlight the significance of the burden of medical services due to diseases that are medically related to COVID-19 (e.g., lung, respiratory, heart, prostate, and diabetes diseases). Moreover, when 1000 persons die of the rest of the possible causes, then about 500 persons die of COVID-19 in the same month (Figure 2C). This may also highlight the significance of the burden of medical services due to medical conditions that are consequences of, for example, the shutdown of the economy, deaths of despair, or murders. In addition, when 1000 persons die of possible causes that are medically related to COVID-19, about 100 persons may die due to the rest of the possible causes in the same month (Figure 2H). Finally, when 1000 persons die due to the rest of the possible causes, then about 100 persons may die by possible causes that are medically related to COVID-19 in the same month (Figure 2F). These may also highlight the importance of the burden of medical services.

Second, we study the dynamic relationships for 12 months after the effects for the t-PQVAR(1) model by aggregating the interaction effects in Figure 2. We found that when 1000 persons die of COVID-19, then about 140 persons may die of deaths that are medically related to COVID-19 (Figure 2D). This relationship may be due to lung, respiratory, heart, prostate, and diabetes diseases (e.g., J44.0, J44.1, J44.9, I27.9, J96.0, J96.1, and J96.9), which may be possibly caused by or whose existing conditions are severely influenced contemporaneously by a COVID-19 infection.

We also found that when 1000 persons die of COVID-19, then about 130 persons may die due to the rest of the possible causes (i.e., non-COVID-19 mortality) (Figure 2G). This positive relationship may be (i) due to panic isolation, (ii) not wanting to burden the US medical services, (iii) no access to health care due to the pressure on US healthcare services (e.g., uninfected patients with Alzheimer’s and Parkinson’s diseases, or other forms of dementia), or (iv) due to the closing of the economy, suicide, or murders.

On the other hand, when 1000 persons die of possible causes that are medically related to COVID-19, about 290 persons may die of COVID-19 (Figure 2B). This positive dynamic relationship may be due to the burden of medical services due to diseases medically related to COVID-19. When 1000 persons die due to the rest of the possible causes, about 230 persons may die of COVID-19 (Figure 2C). This relationship may be due to medical conditions that are consequences of, for example, the shutdown of the economy, deaths of despair, or murders. Moreover, when 1000 persons die of deaths medically related to COVID-19, 115 persons may die due to the rest of the possible causes (Figure 2H). When 1000 persons die due to the rest of the causes, 100 persons may die of causes that are medically related to COVID-19 (Figure 2F). These relationships may also be due to the burden of medical services.

Concerning correlations between possible causes of COVID-19 and non-COVID-19 deaths, our results are consistent with Appleby (2020), Jacobson and Jokela (2020), Woolf et al. (2021, 2020), Bhaskaran et al. (2021), Cronin and Evans (2021), Shiels et al. (2021a, 2021b), and Rosenfeld et al. (2023). We extend them because the t-PQVAR(1) model captures dynamic interaction effects between those variables, which, to our knowledge, are not measured in the literature.

6. Conclusions

In this paper, we have contributed to the literature in the following ways: (i) To the best of our knowledge, this is the first paper that studies the dynamic interaction effects between COVID-19 and non-COVID-19 mortality. (ii) We have used an updated observation period for the causes of death time series concerning other works from the literature. (iii) We have used a novel robust score-driven dynamic panel data model with time-varying unobserved heterogeneity that extends the updating mechanisms of classical PVAR and PVARMA models. We have used a multivariate time series model, PQVAR, to measure dynamic interactions among different possible causes of mortality in the US.

We have used publicly available panel data from the CDC for all US states and the District of Columbia on all possible causes of death. We have used (i) the count of COVID-19 mortality, (ii) the count of deaths that may be medically related to COVID-19, and (iii) the count of other possible causes of death variables for the econometric analysis. Category (ii) includes direct possible causes of COVID-19, for example death due to COVID-19 is registered as deaths due to influenza, pneumonia, or other lung diseases. It also included heart, prostate, and diabetes diseases that are caused by or existing conditions severely influenced by COVID-19. Category (iii) includes indirect causes of COVID-19. Examples of those are not seeking help from medical services due to fear of contracting COVID-19, patients with serious and potentially mortal diseases that are medically unrelated to COVID-19 (such as Alzheimer’s or Parkinson’s disease) being unable to access medical services due to the pressure on healthcare services, or due to the shutdown of the economy, deaths of despair, or murders.

We have shown that t-PQVAR outperforms Gaussian-PVARMA(p,p) and Gaussian-PVAR(p) for the empirical dataset on possible causes of COVID-19 and non-COVID-19 mortality. The conditions of the asymptotic properties of the ML estimator have been supported for the t-PQVAR model. The model diagnostics, model performance metrics, and IRF estimates have clearly shown that the score-driven and information-theoretically optimal t-PQVAR(1) model is superior to the classical Gaussian-PVARMA(1,1), and Gaussian-PVAR(1) models. We have found that the IRFs of t-PQVAR are more precise than the IRFs of Gaussian-PVARMA and Gaussian-PVAR. Therefore, we have discussed the dynamic interactions by focusing on the IRFs of t-PQVAR.

Our results have supported the practical use of the t-PQVAR(1) model and provide robust statistical results on the positive influence of the COVID-19 pandemic on non-COVID-19 causes of death in the US. The estimation and IRF results are US-specific, but the methods can be generalized for other countries or regions. Future work may also use the t-PQVAR(1) model for out-of-sample forecasts of causes of death counts after the outbreak of “Disease X”.