Due to the significant differences in architecture, infrastructure, economy situation among coastal cities, the degree of damage and duration of impact following the landfall of hurricanes are different. At present, the evaluation of urban resilience is mainly based on the selection of resilience evaluation indicators. A large number of indicators (age, real estate value, income etc.) have been identified by the academic community (Stevenson et al., 2010; Olsen et al., 2013; etc.); Tatum et al., 2012), but due to the differences in economic, social and public management systems among countries, the application of existing models and indicators is very limited.
Hurricane is usually considered as external shocking or interruption that pushes a system from operating in equilibrium to a situation of disequilibrium. The existence of common factors shows that these measures are cointegrated or linked for a long time. If the system does not return to a longterm stable equilibrium, then the system is not cointegrated. The government needs to take measures to help the system reach the normal operation of the balanced state. However, in the case of cointegration, the system does return to equilibrium through self adjustment process and shows the ability of resilience.
Stationary tests as the preliminary step for cointegration test are conduct in this study to examine the stationary situation of each variables. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity, trend stationarity or explosive root depending on the test used. A time series is said to be stationary if its statistical properties do not change with time. Augment Dickey Fuller (ADF) test and Phillips and Perron (PP) test are common methods used to examine whether the time series is stationary. In our study, the level of variables is test firstly. If the variable is not stationary in level, then the first difference of variable will be tested to determine the stationary status. A nonstationary series could be stationary in first difference, which is called integrated of order 1.
ADF test includes fitting regression model with ordinary least squares (OLS), but sequence correlation will bring a problem. To illustrate this point, the regression of extended DickeyFuller test includes the lag of the first difference of X. Based on the following equation (Dickey and fuller 1981), the ADF test is performed based on the following equation (Dickey and Fuller 1981):
ΔXt = κ + δt +(ρ − 1)Xt−1 +∑N ψiΔXt−i + υt (1) i=1 (1)
where Xt is the individual time series under investigation, in this study, they are electricity consumption, real estate development investment and total retail sales of consumer goods. Δ is the first difference of the variables, t is a linear time trend, υt is a covariance stationary random error, and the number of lags on the augmenting term N is determined by Akaike's information criterion to ensure sequence uncorrelated residuals. If (ρ − 1) < 1, the testing result is statistically significant at certain level (5% or 1%), then the null hypothesis that X is a nonstationary sequence is rejected.
An alternative unit root test is developed by Phillips and Perron (1988), allowing for weak dependence and heterogeneity in error terms. The proposed method is robust to broad range sequence correlation and timedependent heteroscedasticity. PP tests for unit root is generally expressed as:
X t = ξ0 + ξ1(t − T2) + λXt−1 + γt (2)
where (tT/2) is the time trend, T is the sample size, and γt is the error term.
In contrast to the alternative hypothesis that Xt is stationary around the determined trend (Ha: λ < 1), null hypothesis of the unit root Ho: λ = 1 is tested. MacKinnon (1991) critical values is applied to determine the statistical significance of ADF test and PP test.
Table 4 Unit Root Tests of Three Variables in Xiamen
Notes: E denotes the log of electricity consumption, R denotes the log of real estate development investment and S denotes the log of total retail sales of consumer goods. Δ denotes firstdifference operator. A*denotes statistically significant at the one percent level or less based on MacKinnon (1991) critical values.
Table 5 Unit Root Tests of Three Variables in Fuzhou
The results of the ADF and PP unit root tests in Xiamen and Fuzhou are presented in Table 4 and Table 5 respectively and suggest that each variable is nonstationary in levels. The impact on one of the series alone is permanent because the series will not return to its longterm average over time. However, the first difference of each time series is stationary. As mentioned above, it is possible that a linear combination of nonstationary I (1) variables in levels may be stationary. In this study, these variables are called cointegration variables, although each variable may not return to a specific value alone for a longterm trend, there is a common factor that leads the group (consisting of two or more variables) to have a common tendency to converge in the long term. Given the results of the ADF and PP tests, a group test called cointegration test is proceeded to examine the cointegrating properties of the variables.
Cointegration Test
The cointegration test analyzes a nonstationary time series process, the variance and mean of the process change with time. In other words, cointegration is a test method used to examine the possible longterm comovement among time series variables. The most popular cointegration tests are EngleGranger and Johansen Test. If the linear combination of two or more variables has a lower order of integration, they will be cointegrated. For example, if a linear combination of I (0) can be used to model a set of I (1) variables, there is cointegration. The order of integration I (1) reveal that a set of differences can transform a nonstationary variable into stationarity. If two or more variables are cointegrated together, a common factor is existed behind the variables, and the system could return to equilibrium without external adjustment, which could prove the ability of resilience. If the system could not return to longrun equilibrium, government action or assistance are needed to help the affected area return to normal operation state.
The maximum likelihood method of Johansen (1988) and Johansen and Juselius (1990) is applied in this study to determine the number of cointegrating vectors. The testing process is based on the Porder vector autoregression (VAR) model of order p:
$$\varDelta {X}_{i}=\mu +{{\Gamma }}_{1}{\Delta }{X}_{t1}+\dots +{{\Gamma }}_{k}{\Delta }{X}_{tk1}{\Pi }{X}_{tk}+{\epsilon }_{t}$$
3
Where
$${{\Gamma }}_{i}=I+{{\Pi }}_{1}+\dots +{{\Pi }}_{i}$$
4
And
$${\Pi }=I{{\Pi }}_{1}\dots {{\Pi }}_{k}$$
5
Where Xt is the vector of the nonstationary I(1) variable under study, and εt is the independent Gaussian variable(zero mean and Ω variance) in k dimension. The vector Γi contains shortterm parameters that capture the imbalance characteristics of the data. The Π (k × k) matrix includes the information related to longterm relationships among the three variables under the examination.
The cointegration test determines the rank of the matrix Π. In order to determine the rank, two tests were carried out, namely the trace test and the maximum eigenvalue test. For both of the test statistics, the initial Johansen test is a test of the null hypothesis that there is no cointegration. The forms of alternative hypotheses are various.
The maximum eigenvalue test starts with the test of rank (Π) = 0 and the alternative hypothesis is that rank (Π) = 1. If the rank of the matrix is zero, the maximum eigenvalue is zero, there is no cointegration, and the test ends. If the maximum eigenvalue λ1 is not zero, the rank of the matrix is at least one, in other words, more cointegration vectors may exist. Then test whether the second largest eigenvalue λ2 is zero. If the eigenvalue is zero, the tests are completed, and there is only one cointegration vector. If the second largest eigenvalue λ2 = 0, there are more than two variables, and more cointegration vectors may exist. The testing process continues until the null hypothesis that the eigenvalue is equal to zero cannot be rejected. In summary, the maximum eigenvalue test first tests whether the largest eigenvalue is zero relative to the alternative that the next largest eigenvalue is zero.
The test statistic is as follows:
$$LR\left({r}_{0}, {r}_{0}+1\right)=Tln(1{\lambda }_{r0+1})$$
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Where LR (r0, r0 + 1) is the likelihood ratio test statistic, used to test the null hypothesis of rank (Π) = r0 and the alternative hypothesis of rank (Π) = r0 + 1. For example, the likelihood ratio test statistic LR (0,1) = T ln(1λ1) is used to test the hypothesis of rank (Π) = 0 relative to rank (Π) = 1.
Trace Test
The trace test is to test whether the rank of the matrix Π equals to r0. The null hypothesis is that rank(Π) = r0. The alternative hypothesis is that r0 < rank(Π) ≤ n, where n is the maximum number of possible cointegrating vectors(reference). If the null hypothesis is rejected, the next null hypothesis is rank (Π) = r0 + 1, and the alternative hypothesis is r0 + 1 < rank (Π) ≤ n. For the subsequent tests, it repeats the testing process above, the testing value plus one each time, until it reaches the number n.
The likelihood ratio test statistic is
$$LR\left({r}_{0},n\right)=T\sum _{i={r}_{0}+1}^{n}\text{l}\text{n}(1{\lambda }_{i})$$
7
Where \(LR({r}_{0},n)\) is the likelihood ratio statistic. For example, the likelihood ratio test statistic \(LR\left(0,n\right)=T \sum _{i=1}^{n}\text{l}\text{n}(1{\lambda }_{i})\) is applied to test the null hypothesis versus the alternative that rank(Π) ≤ n.
Johansen and Juselius (1990) and OsterwaldLenum (1992) provide critical values for these two tests. A critical value is used to determine whether to reject the null hypothesis. If the absolute value of the test statistic is greater than the critical value, you can reject the null hypothesis. Table 6  Table 9 report results for the cointegration tests. The alternative hypothesis of having a long run cointegration relationship among three time series is accepted. The results indicate that there is exactly one cointegrating vector among the three analyzed variables in city Fuzhou and Xiamen.
Table 6
Unrestricted Cointegration Rank Test (Trace) in Fuzhou
Hypothesized

Eigenvalue

Trace

Critical Value at 5%

Prob.**

None*

0.165008

35.66217

29.79707

0.0094**

At most 1

0.084002

12.94023

15.49471

0.1170

At most 2

0.014848

1.884886

3.841466

0.1698

Trace test indicates 1 cointegrating at the 0.05 level
Table 7
Unrestricted Cointegration Rank Test (Maximum Eigenvalue) in Fuzhou
Hypothesized

Eigenvalue

MaxEigen
Statistic

Critical Value at 5%

Prob.**

None*

0.165008

22.72194

21.13162

0.0296**

At most 1

0.084002

11.05534

14.26460

0.1515

At most 2

0.014848

1.884886

3.841466

0.1698

MaxEigenvalue test indicates 1 cointegrating at the 0.05 level
Table 8
Unrestricted Cointegration Rank Test (Trace) in Xiamen
Hypothesized

Eigenvalue

Trace

Critical Value at 5%

Prob.**

None*

0.165077

34.72380

29.79707

0.0125**

At most 1

0.070924

11.99147

15.49471

0.1573

At most 2

0.021374

2.722376

3.841465

0.0989

Trace test indicates 1 cointegrating at the 0.05 level
Table 9
Unrestricted Cointegration Rank Test (Maximum Eigenvalue) in Xiamen
Hypothesized

Eigenvalue

MaxEigen
Statistic

Critical Value at 5%

Prob.**

None*

0.165077

22.73232

21.13162

0.0295**

At most 1

0.070924

9.269095

14.26460

0.2644

At most2

0.021374

2.722376

3.841466

0.0989

MaxEigenvalue test indicates 1 cointegrating at the 0.05 level
Construct Index
In this section, the composite indicator construction method recorded in the Conference Board (2001) will be used to construct the index to evaluate the resilience of affected city based on the selected variables in the last section. Its components include real estate development investment, electricity consumption and retail sales of consumer goods at the city level. The composite index takes into account the built environment, the ability of customer purchasing and energy distribution, thus representing the disaster resilience capacity of the given community. The process of calculating composite index involves transforming the indicators to similar ranges, aggregating the time series indicators in a composite form and recalculating the composite index in the base year. The aggregation procedure can be described in detail in the following 5 steps:
In the first step, monthly changes are computed for each component over the entire sample period. The subscript of t and t1 represent the current month and last month, respectively, and m represent specific components of the index. If the component is not in the form of percentage change, the symmetry alternative could be used to substitute the traditional percentage change formula:
$${r}_{i,t}= \frac{200\times \left({X}_{i,t}{X}_{i,t1}\right)}{\left({X}_{i,t}+{X}_{i, t1}\right)}$$
8
where, \({r}_{i,t}\) is monthtomonth changes, \({X}_{i,t}\) is time series data for component \(i\)at month \(t\), in our case, \({X}_{i,t}\) is electricity consumption, real estate development investment and total retail sales of consumer goods.
In the second step, the monthly changes are adjusted to balance the volatility of each component. Standard deviations of the changes in each component are computed, inverted, and their sum are also obtained. The adjusted monthly change in each component is multiplied by the corresponding standardization factors.
$${w}_{i,year}= \frac{{\frac{1}{std}}_{i,year}}{{\sum }_{1}^{n}\left({\frac{1}{std}}_{i,year}\right)}$$
9
Where \(n\) is the number of components, and standardization factors are updated annually and consistent with the annual benchmark revision of the Conference Board.
The third step is comprised of the adjusted monthly contributions of each component are added up to obtain the growth rate of the index. The sum of adjusted contributions can be obtained in this step, that is, the monthly growth rate of the index.
$${c}_{i,t}={w}_{i,year}\times {r}_{i,t}$$
10
where \({c}_{i,t}\) is monthly contribution for each component (The Conference Board 2001).
In the fourth step, the sum of the growth rate of the composite index is adjusted so that its trend is similar to coincident index. This is achieved by adding an adjustment factor to the monthly growth rate of the index. For example, the trend adjustment factor of the leading index is calculated by subtracting its average monthly growth rate from the average monthly growth rate of the coincident index. The symmetric percentage change formula is used to calculate the index level in the fifth step. The index is calculated from the initial value of 100 in the first month of the sample period. In the final step, the index will be readjusted to the average of 100 in 2010, the historical index is multiplied by 100, and then divided by the 12month average in 2010. \({s}_{t}\) is the sum of the monthly contributions across the component, and \(n\) is the number of components, the number is three in our study.
$${s}_{t}=\sum _{i=1}^{n}{c}_{i,t}$$
11
These annually updated standardized factors should be consistent with the annual benchmark of the Conference Board. The third step is to assign the standardized coefficient wi and monthtomonth changes ri;t to each component. The monthly contribution of each component is calculated as follows: where ci;t is monthly contributions of each component (The Conference Board 2001).
In the next step, the initial index is calculated recursively based on the symmetrical percentage change formula, starting from 100 in the first month of the sample.
$${I}_{j}={I}_{j1}\times \frac{(200+{s}_{j})}{(200{s}_{j})} (j=\text{2,3},\dots ,m)$$
12
where \(m\) is the total number of month in the sample period. The results of Typhoon Resilience Index in Xiamen and Fuzhou are shown in Figures below.