Id |
Subject |
Object |
Predicate |
Lexical cue |
T868 |
0-11 |
Sentence |
denotes |
Appendix C. |
T869 |
12-42 |
Sentence |
denotes |
Computation of counterfactuals |
T870 |
43-60 |
Sentence |
denotes |
Our main model is |
T871 |
61-195 |
Sentence |
denotes |
4 yct=∑τ=12∑k=1Kwithinαwithin,τkh¯ctkτy¯ctτ+∑τ=12∑k=1Kbetween∑r≠cαbetween,τkm¯crtkτy¯rtτ+∑τ=12∑k=1KWuhanρτkm¯c,Wuhan,tkτz¯tτ+xctβ+𝜖ct. |
T872 |
196-311 |
Sentence |
denotes |
It is convenient to write it in vector form, 5 Ynt=∑s=114Hnt,s(αwithin)+Mnt,s(αbetween)Yn,t−s+∑τ=12Zntτρτ+Xntβ+𝜖nt, |
T873 |
312-357 |
Sentence |
denotes |
where Ynt=y1t⋯ynt′ and 𝜖nt are n × 1 vectors. |
T874 |
358-519 |
Sentence |
denotes |
Assuming that Yns = 0 if s ≤ 0, because our sample starts on January 19, and no laboratory confirmed case was reported before January 19 in cities outside Wuhan. |
T875 |
520-579 |
Sentence |
denotes |
Xnt=x1t′⋯xnt′′ is an n × k matrix of the control variables. |
T876 |
580-716 |
Sentence |
denotes |
Hnt,s(αwithin) is an n × n diagonal matrix corresponding to the s-day time lag, with parameters αwithin={αwithin,τk}k=1,⋯,Kwithin,τ=1,2. |
T877 |
717-928 |
Sentence |
denotes |
For example, for s = 1,⋯ , 7, the i th diagonal element of Hnt,s(αwithin) is 17∑k=1Kwithinαwithin,1kh¯ct,ik1, and for s = 8,⋯ , 14, the i th diagonal element of Hnt,s(αwithin) is 17∑k=1Kwithinαwithin,2kh¯ct,ik2. |
T878 |
929-970 |
Sentence |
denotes |
Mnt,s(αbetween) is constructed similarly. |
T879 |
971-1082 |
Sentence |
denotes |
For example, for s = 1,⋯ , 7 and i≠j, the ij th element of Mnt,s(αbetween) is 17∑k=1Kbetweenαbetween,1km¯ijtk1. |
T880 |
1083-1157 |
Sentence |
denotes |
Zntτ is an n × KWuhan matrix corresponding to the transmission from Wuhan. |
T881 |
1158-1218 |
Sentence |
denotes |
For example, the ik th element of Znt1 is m¯i,Wuhan,tk1z¯t1. |
T882 |
1219-1306 |
Sentence |
denotes |
We first estimate the parameters in Eq. 4 by 2SLS and obtain the residuals 𝜖^n1,⋯,𝜖^nT. |
T883 |
1307-1396 |
Sentence |
denotes |
Let ⋅^ denote the estimated value of parameters and ⋅~ denote the counterfactual changes. |
T884 |
1397-1639 |
Sentence |
denotes |
The counterfactual value of Ynt is computed recursively, Y~n1=∑τ=12Z~n1τρ^τ+Xn1β^+𝜖^n1,Y~n2=∑s=11H~n2,s(α^within)+M~n2,s(α^between)Y~n,2−s+∑τ=12Z~n2τρ^τ+Xn2β^+𝜖^n2,Y~n3=∑s=12H~n3,s(α^within)+M~n3,s(α^between)Y~n,3−s+∑τ=12Z~n3τρ^τ+Xn3β^+𝜖^n3,⋮ |
T885 |
1640-1694 |
Sentence |
denotes |
The counterfactual change for date t is ΔYnt=Y~nt−Ynt. |
T886 |
1695-1765 |
Sentence |
denotes |
The standard error of ΔYnt is obtained from 1000 bootstrap iterations. |
T887 |
1766-1883 |
Sentence |
denotes |
In each bootstrap iteration, cities are sampled with replacement and the model is estimated to obtain the parameters. |
T888 |
1884-2051 |
Sentence |
denotes |
The counterfactual predictions are obtained using the above equations with the estimated parameters and the counterfactual scenario (e.g., no cities adopted lockdown). |