We begin this section with the discussion of the fractional persistence results when the time trend is assumed to follow a linear pattern. As conventional in the literature, the fractional integration model is estimated under three different assumptions for the deterministic terms, namely model with no deterministic terms, model with only a constant (intercept), and model with both constant and a linear trend. As clearly observed in Table 2, the model with constant and linear trend seems to produce the estimates for the global sample and virtually all the countries. The only exemption is China which favours the model with a constant. Table 3 thus represents the results of the best models form Table 2. For the global estimate, the unit root hypothesis is resoundingly rejected in favour of I(d>1), thus indicating a high degree of persistence. Turning to the countryspecific cases, the same conclusion seems appropriate for China whose fractional differencing coefficient is given as 1.3654. Also, the rejection of the I(1) null hypothesis is ascertained for three other countries, namely France, United Kingdom and United States, although it is in respect of 0.5 ≤ d < 1, signifying nonstationary mean reversion. Only in the cases of Italy and Spain can we say mortality rates exhibit random walk since the d estimates are closer to unity.
Table 2: Fractional persistence with linear trend
Series

No regressors

An intercept

A linear time trend

World estimate

1.0415***
(0.0307)

1.3625***
(0.0793)

1.2582***
(0.0897)

Countryspecific estimates

China

1.0361***
(0.0328)

1.3654***
(0.0819)

1.3302***
(0.0845)

France

1.1072***
(0.0514)

1.2714***
(0.0867)

0.8881***
(0.1291)

Italy

0.8463***
(0.0416)

0.9998***
(0.0003)

0.9751***
(0.0836)

Spain

0.9490***
(0.0667)

0.9999***
(0.0005)

0.9673***
(0.1451)

United Kingdom

0.9928***
(0.0754)

1.0000***
(0.0005)

0.6795***
(0.1777)

United States

1.0527***
(0.0557)

1.3545***
(0.1020)

0.7985***
(0.1901)

*** represents significance at 1% critical level. Estimated values in bold correspond to the best model, while those in parentheses are the standard errors.
Table 3: Results of the best models from Table 2
Series


Intercept

Trend

World estimate

1.2582***
(0.0897)

8.1894***
[11.70]

0.2033***
[2.68]

Countryspecific estimates

China

1.3654***
(0.0819)

6.7053***
[12.90]



France

0.8881***
(0.1291)

13.6763***
[2.77]

0.2899***
[8.43]

Italy

0.9751***
(0.0836)

54.0793
[0.26]

0.1056**
[2.11]

Spain

0.9673***
(0.1451)

8.5626
[0.09]

0.2562***
[2.86]

United Kingdom

0.6795***
(0.1777)

12.1119***
[6.92]

0.3235***
[8.46]

United States

0.7985***
(0.1901)

12.3337***
[9.19]

0.2826***
[11.60]

*** represents significance at 1% critical level. Estimated values in parentheses and brackets are respectively the standard errors and tvalues.
However, certain empirical studies have challenged the modeling of the degree of persistence in mortality rates with linear models. In particular, they argue that mortality rates exhibit nonlinear dynamics (see Hill et al., 1999; Booth et al., 2002; Shang et al., 2006, Yaya and GilAlana, 2018). Incorporating this tendency, we further use the approach of Cuestas and GilAlana (2016) to account for nonlinearities along the time trend of the mortality rates of our sampled cases. The approach is flexible is that it helps to determine different nonlinear orders, such that significance at any polynomial degree from 2 symbolizes nonlinearity along the path of the series. However, we restrict the polynomial degree to 3 as conventional in the literature (see Yaya and GilAlana, 2018), as that is enough to infer the presence of nonlinearities. In light of this, we start with the estimation of the nonlinear model when secondorder polynomial is assumed. The results are presented in Table 4. The only country whose mortality rates do not follow a nonlinear trend is the United Kingdom. Except for the global, China and France mortality rates whose results are consistent with the linear case, significant changes are observed in the d estimates of the remaining countries. While the situation becomes stationary mean reverting for the United States due to its value being lower than 0.5 (i.e. 0.3496), the estimates of Italy and Spain turn negative, implying short memory or antipersistence.
We extend the nonlinearity focus to polynomial of order 3. This higher order is important it provides a more accurate d estimate if the thirdorder time coefficient is significant. In what seems to affirm this assertion, Table 5 shows that for the cases where the thirdorder time coefficients and d estimates are significant (world, China, Italy and Spain), there are alterations to the results reported above. The findings suggest that at the global scene and in China, mortality rates are nonstationary mean reverting since their estimates are now lower than 1, but above 0.5, but shortmemory is evident for Italy and Spain following their negative estimates. As a confirmation, such evidence is also found in Table 4 for Italy and Spain. For the other countries whose d estimates are not significant, it means the nonlinearities in the time trend of their mortality rates are only inherent to the second degree, except the United Kingdom, thus making us to rely still on the results with m=2 in Table 4.
Following from these results, we establish that nonlinearities matter in the time trends of the mortality rates of the global and countrybased samples, except the United Kingdom. So, mean reversion is evident for the world, China, France, the United Kingdom and the United States, although it is faster for the United States, followed by France. This implies that the effect of shocks as caused by the outbreak of the COVID19 is temporary. In other words, the mortality rates in these countries are not induced by factors that warrant intense policies to be formulated to restore normalcy. Even with the present social and economic policies adopted by these countries, mortality rates will still revert back to their original trend in no distant time, regardless of the high number recorded in some of the countries, especially United States. On the other hand, mortality rates in Italy and Spain exhibit short memory, i.e. they are antipersistent.
Table 4: Nonlinear Fractional Persistence based on Chebyshev inequality with n=2
Series





World estimate

1.2558***
(0.0884)

1.2427
[0.50]

3.2097**
[2.24]

1.0714*
[1.93]

Countryspecific estimates

China

1.1430***
(0.0941)

0.0532
[0.03]

2.8187***
[3.21]

1.6047***
[4.22]

France

0.5031***
(0.1441)

1.6613***
[6.36]

3.7298***
[15.60]

0.8959***
[5.87]

Italy

0.3597**
(0.1320)

2.7216***
[73.20]

4.6614***
[106.00]

0.7538***
[25.90]

Spain

1.1014***
(0.2756)

6.3405***
[145.00]

7.9188***
[163.00]

1.2749***
[46.70]

United Kingdom

0.1212
(0.2283)

4.5984***
[7.62]

5.4600***
[8.16]

0.1694
[0.58]

United States

0.3496*
(0.2006)

0.1110
[0.22]

0.3644
[0.71]

2.3508***
[11.10]

***, ** and * represent significance at 1%, 5% and 10% critical levels respectively. Values in parentheses and brackets are respectively standard errors and tvalues, while the bolded fractional differencing coefficients denote the significance of nonlinear time trend at the secondorder polynomial.
Table 5: Nonlinear Fractional Persistence based on Chebyshev inequality with n=3
Series






World estimate

0.6385***
(0.1233)

1.1850***
[3.29]

3.8330***
[26.80]

1.1378***
[11.80]

1.0636***
[14.2]

Countryspecific estimates

China

0.8149***
(0.1105)

1.8600
[1.54]

3.0472***
[12.00]

1.7223***
[11.50]

0.7019***
[6.39]

France

0.2673
(0.1626)

0.9847***
[4.55]

2.9736***
[11.70]

1.4316***
[7.77]

0.3573***
[3.46]

Italy

0.5961***
(0.1985)

2.8659***
[38.10]

4.8457***
[49.90]

0.8796***
[13.10]

0.0591*
[1.85]

Spain

1.4310***
(0.2863)

6.7793***
[42.30]

8.4562***
[43.20]

1.5963***
[13.70]

0.1232**
[2.81]

United Kingdom

0.2077
(0.2608)

9.7047***
[5.93]

11.5026***
[5.93]

3.2613***
[3.01]

1.0993***
[3.23]

United States

0.3020
(0.2768)

4.3772***
[6.01]

5.2811***
[6.15]

0.3242
[0.68]

0.8008***
[5.47]

***, ** and * represent significance at 1%, 5% and 10% critical levels respectively. Values in parentheses and brackets are respectively standard errors and tvalues, while the bolded fractional differencing coefficients denote the significance of nonlinear time trend at the thirdorder polynomial.