Online Appendix Validating China s output data using satellite observations

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1 Online Appendix Validating China s output data using satellite observations Stephen D. Morris Department of Economics Bowdoin College Junjie Zhang Duke Kunshan University and Duke University February 2, 21 Revised: September 2, 21 Bowdoin College, Department of Economics, 9 College Station, Brunswick, ME 411. smorris@bowdoin.edu. Duke Kunshan University, Environmental Research Center; Duke University, Nicholas School of the Environment. No. Duke Avenue, Kunshan, Jiangsu, China junjie.zhang@duke.edu. 1

2 A Supplementary tables and figures Figure A.1: Reported GDP vs. electricity generation, Great Recession period. China Beijing Tianjin Hebei Shanxi Inner Mongolia Liaoning Jilin Heilongjiang Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong Henan Hubei Hunan Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Shaanxi Gansu Qinghai Ningxia Xinjiang Notes: Units are equivalent to Figure 1 (a) in main paper (All-China: Top-left pane in this Figure). Scale varies by location. National level and N = 31 sub-national regions: Quarterly four-quarter rolling average, Shaded: NBER recession dates (Q4-9Q2). Black inset: GDP NBER dates % change. Red inset: Electricity NBER dates % change.

3 Figure A.2: Reported GDP vs. NO x emissions, Great Recession period. China Beijing Tianjin Hebei Shanxi Inner Mongolia Liaoning Jilin Heilongjiang Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong Henan Hubei Hunan Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Shaanxi Gansu Qinghai Ningxia Xinjiang Notes: Units are equivalent to Figure 1 (b) in main paper (All-China: Top-left pane in this Figure). Scale varies by location. National level and N = 31 sub-national regions: Quarterly four-quarter rolling average, Shaded: NBER recession dates (Q4-9Q2). Black inset: GDP NBER dates % change. Blue inset: NO x NBER dates % change. 3

4 Figure A.3: Annual % change, China: -. (a) Satellite-measured signals. 3 2 GDP Luminosity NOx (b) Reported signals GDP Freight Electricity Cement Steel Notes: Shading: Asian Financial Crisis. 4

5 Figure A.4: Annualized quarterly % change, China: 26 Q1-213 Q3. (a) Satellite-measured signals. 3 3 GDP NO Q1 Q1 Q1 9 Q1 1 Q1 11 Q1 12 Q1 13 (b) Reported signals. 6 4 GDP Electricity Cement Steel Q1 Q1 Q1 9 Q1 1 Q1 11 Q1 12 Q1 13 Notes: Shading: NBER recession dates.

6 Figure A.: Annual elasticities by-region (-)..3 ˆβ =.4 China 3.1 ˆβ =.2 Beijing 3.2 ˆβ =.3 Tianjin 1.9 ˆβ =.4 Hebei ˆβ =. ˆβ =.2 ˆβ =.4 ˆβ =. Shanxi Inner Mongolia Liaoning.3 Jilin ˆβ =.3 ˆβ =. ˆβ =.4 ˆβ =.4 ˆβ =. ˆβ =.6 ˆβ =.4 ˆβ =.4 Heilongjiang Shanghai Jiangsu 6. Zhejiang ˆβ =. ˆβ =.3 ˆβ =. ˆβ =.6 ˆβ =.3 ˆβ =.4 ˆβ =.6 ˆβ =. Anhui Fujian Jiangxi. Shandong ˆβ =. ˆβ =. ˆβ =. ˆβ =.4 ˆβ =.6 ˆβ =. ˆβ =.4 ˆβ =. Henan Hubei Hunan. Guangdong ˆβ =. ˆβ =. ˆβ =. ˆβ =.3 ˆβ =.6 ˆβ =. ˆβ =.4 ˆβ =. Guangxi Hainan Chongqing.1 Sichuan ˆβ =.6 ˆβ =. ˆβ =.9 ˆβ =. ˆβ =. ˆβ =. ˆβ =. ˆβ =. Guizhou Yunnan Tibet.2 Shaanxi ˆβ =. ˆβ =. ˆβ =.9 ˆβ =.6 ˆβ =. ˆβ =.6 ˆβ =.4 ˆβ =.4 Gansu Qinghai Ningxia 6.4 Xinjiang ˆβ =. ˆβ =. ˆβ =. ˆβ =. ˆβ =. ˆβ =. ˆβ = ˆβ = Notes: Pink: Luminosity. Blue: NO x emissions. Across T = 16 quarters in each pane. Estimates are for regression model ln Signal t = β ln Output t + ε t for each n. 6

7 Figure A.6: NO 2 quarterly elasticities by-region. 6. China 4 Beijing 4.3 Tianjin 3.9 Hebei ˆβ = Shanxi ˆβ = InnerMongolia ˆβ = Liaoning ˆβ =. 2.. Jilin ˆβ = Heilongjiang ˆβ = Shanghai ˆβ = Jiangsu ˆβ = Zhejiang.2 3. ˆβ = Anhui.3 3. ˆβ = Fujian. 2.2 ˆβ = Jiangxi ˆβ = Shandong ˆβ = Henan. 3. ˆβ =.2 1. Hubei. 3 ˆβ = Hunan ˆβ = Guangdong ˆβ = Guangxi ˆβ = Hainan. 1.9 ˆβ = Chongqing. 2. ˆβ = Sichuan ˆβ = Guizhou ˆβ = Yunnan ˆβ = Tibet.1 1. ˆβ = Shaanxi.9 2. ˆβ = Gansu.6 2 ˆβ = Qinghai 2 ˆβ = Ningxia.4 2. ˆβ = Xinjiang ˆβ = ˆβ = ˆβ = ˆβ = Notes: Across T = 31 quarters in each pane. Estimates are for regression model ln Signal t = β ln Output t + ε t for each n.

8 Figure A.: Elasticities by-year ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =1.3 ˆβ =1.3 ˆβ =1.2 ˆβ = ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =1.2 ˆβ =1.2 ˆβ = ˆβ = ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =. ˆβ =1.2 ˆβ =1.2 ˆβ =1.2 ˆβ = ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =.9 ˆβ =1.2 ˆβ =1.2 ˆβ =1.2 ˆβ = Notes: Pink: Luminosity. Blue: NO x emissions. Across N = 31 regions in each pane. Estimates are for regression model ln Signal n = β ln Output n + ε n for each t.

9 Figure A.: NO 2 elasticities by-quarter Q1 6 2 Q2. 2 Q Q Q1 1.3 ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ = Q Q Q Q Q ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ = Q Q Q Q Q ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ = Q Q Q Q Q ˆβ =.61.3 ˆβ =.61.2 ˆβ =.61.2 ˆβ =.61.2 ˆβ = Q Q Q Q Q ˆβ =.1.2 ˆβ =.1.3 ˆβ =.1.2 ˆβ =.1.2 ˆβ = Q Q Q Q Q ˆβ =.1.2 ˆβ =.1.2 ˆβ =.1.3 ˆβ =.1.3 ˆβ = Q Q Q Q Q ˆβ =.61.3 ˆβ =.1.3 ˆβ =.1.3 ˆβ =.1.2 ˆβ = Notes: Across N = 31 regions in each pane. Estimates are for regression model ln Signal n = β ln Output n + ε n for each t. 9

10 Table A.1: Correlations, annual sample (hundredths). GL GN GF GE GC GS LN LF LE LC LS NF NE NC NS China Beijing Tianjin Hebei Shanxi In. Mon Liaoning Jilin Heil Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong Henan Hubei Hunan Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet NA NA NA Shaanxi Gansu Qinghai Ningxia Xinjiang Mean Max Count Notes: -. G: GDP. L: Luminosity. N: NO x. F: Freight. E: Electricity. C: Cement. S: Steel. Max count gives number of areas for which the leading G, L, or N correlation is the largest (bold). 1

11 Table A.2: Correlations, quarterly sample (hundredths). GN GE GC GS NE NC NS China Beijing Tianjin Hebei Shanxi Inner Mon Liaoning Jilin Heilongjiang Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong Henan Hubei Hunan Guangdong Guangxi Hainan NA 1 9 NA Chongqing Sichuan Guizhou Yunnan Tibet NA NA Shaanxi Gansu Qinghai Ningxia NA 1-41 NA Xinjiang Mean Max Count Notes: 26Q1-213Q3. G: GDP. L: N: NO 2. E: Electricity. C: Cement. S: Steel. Max count gives number of areas for which the leading G or N correlation is the largest (bold). 11

12 Figure A.9: China regional signal vs. GDP change: NBER recession dates (2 Q4-29 Q2). 12 GDP NBER dates change (%) Guangdong Ningxia Heilongjiang Xinjiang Inner Mongolia Heilongjiang Electricity NO2 Anhui NO2 / Electricity NBER dates change (%)

13 B Output This section justifies the assumption that ynt has an AR(1) rule of motion (2) using standard macroeconomic theory. Consider a dynamic stochastic general equilibrium model not explicitly derived from microfoundations, but incorporating the main features of a wide class of models. It consists of a Taylor rule, linearized Euler equation, and Phillips curve. r nt = ψ π π nt + ψ y y nt + ε r nt (B.1) c nt = E t c nt+1 + hc nt 1 (1/τ)(r nt E t π nt+1) (B.2) π nt = βe t π nt+1 + κy nt (B.3) rnt is the nominal interest rate, c nt is nominal consumption, and πnt is inflation. Each is stated in log difference from means, and with-stars to denote they are true, possibly unobservable values. Inasmuch they are directly comparable with ynt. τ is the coefficient of relative risk aversion, h is a parameter governing the degree of habit formation, β is the discount factor, and κ is a function of the degree of price stickiness. ψ π and ψ y are indicative of monetary policy stance. ε r nt IW N(, 1). With no investment or government spending, we have the market-clearing condition c nt = ynt. Since c nt 1 is the only lagged 13

14 variable in the system, the solution will have the form of a restricted singular VAR(1). y nt r nt π nt = ρ y ρ r ρ π ynt 1 + σ y σ r σ π ε r nt (B.4) The VAR parameters are nonlinear functions of the underlying parameters of the model. This fact may be confirmed using the method of undetermined coefficients. Restricting the set of solutions to those for which ρ y (, 1), the first row of this VAR is simply Equation (2). But possibly many models have this same VAR solution. So, this particular structural model is not the unique underpinning of the assumption that ynt follows an AR(1). Finally, if ε y nt is interpretable as a scaled monetary policy shock σ y ε r nt, then the shocks should reasonably be correlated across areas n in a country. This is allowed by the assumptions. C Representation (Proof of Proposition 1) The model (2)-(6) may be stated in by-area n ABCD state space representation common of many macroeconomic models, for each signal i (Fernández- 14

15 Villaverde et al., 2). y nt u nt y nt = s nt (i) } {{ } Y nt(i) = ρ u } {{ } A ρ y ρ u β(i)ρ y } {{ } C(i) y nt 1 u nt 1 y nt 1 u nt } {{ } B 1 1 β(i) 1 } {{ } D(i) ε nt (i) ε y nt ε u nt ε s nt(i) }{{} ε nt(i) (C.1) The admissible parameter space Θ(i) for θ(i), the structural parameters corresponding to any lone signal i, has been defined from economic fundamentals and benign normalizations. Θ(i) : β(i), < ρ y < 1, < ρ u < 1, < σ y <, < σ u <, < σ(i) < When θ(i) Θ(i) the matrix C(i) is upper-triangular and thus always invertible. So the observation equation implies [y nt 1 u nt 1] = C(i) 1 Y nt (i) C 1 (i)d(i)ε nt (i) and thus Y nt (i) = ρ u ψ(i) Y nt 1(i) + vnt(i) y ρ y vnt(i) s (C.2) 1

16 vnt(i) y = D(i)ε nt(i) + ψ(i) ε nt 1 (i) = u y nt(i) + ψ(i)m(i) vnt(i) s ρ y u s nt(i) ρ y m(i) }{{}}{{} C(B AC 1 D) D(i)ε nt(i) ψ(i) = λ i ρ y ρ u β(i) m(i) = σ(i) β(i) 2 σy 2 + σ 2 (i) u y nt 1(i) u s nt 1(i) (C.3) This is VARMA(1,1) representation. See Morris (216) for details on VARMA representations of ABCD models. Now consider the general case of S 1 signals. Given (C.2), y nt ρ u ψ(1)... ψ(s) y S nt 1 i=1 λ ivnt(i) y s nt (1) ρ y... s nt 1 (1) v s = + nt(1) s nt (S)... ρ y s nt 1 (S) vnt(s) s }{{}}{{}}{{} Y nt P V nt (C.4) for any weightings satisfying S i=1 λ i = 1. Thus, λ = [λ 1,... λ S ] are nuisance parameters for all S > 1. They are only known when S = 1 (when λ 1 = 1). Using (C.3), the generalized error is V nt = S i=1 λ ivnt(i) y v s nt S 1 = S ε nt + β I s S 1 S 1 }{{} M (S+1) (S+2) ψ 1 S I s ρ y S 1 S 1 }{{} M 1 (S+1) (S+2) ε nt 1 (C.) 16

17 for ψ = (λ 1 ψ(1),..., λ S ψ(s)) and ε nt the multiple signal generalization of ε nt (i). [ ] ε nt = ε y nt ε u nt ε s ; Σ nt ε = E(ε ntε nt) = (S+2) 1 σy 2 σu 2 Σ (C.6) Next, using elementary matrix algebra operations, we have the equivalence P Y nt 1 = [ ])] vec(p Y nt 1 ) = (Y nt 1 I S+1 )vec(p ) where vec(p ) = ρ u 1 S K S,S+1 vec ([ψ I s ρ y with K S,S+1 is the S(S + 1)-dimensional square commutation matrix for which we have ]) [ ] the exact equality K S,S+1 vec ([ψ I s ρ y = vec( ψ I s ρ y ) for ψ defined by (1) (Abadir and Magnus, 2). Thus, vec(p ) = RΨ for R the zero-one selection matrix, R = (S+1) 2 (S+2) I S+1 S(S+1) (S+1) K S,S+1 (S+1) (S+1) I S S 1 S 2 S 1 1 S S 1 S 1 S S I S S 1 S 1 vec(i S ) 1 1 S (C.) Inasmuch, (C.4) may be exactly restated in the form of (16) in the text. For future use, (C.4) additionally has the Wold decomposition Y nt = M ε nt + P i (M 1 + P M ) ε nt i 1 i= (C.) 1

18 Finally, we must compute the variance-covariance matrix E(V nt V nt) = Ω. Using (C.), Ω = M Σ ε M + M 1 Σ ε M 1 (C.9) D Identification and estimation of reduced form (Proof of Proposition 2) Here and henceforth we consider the case without a time decay in the signal-output relationship (Appendix H above). We wish to establish consistency and asymptotic normality of Ψ (Equation (21)). The model (16) yields by-n representation Y n = [ ] [ X n Ψ + V n with instruments Z n for Y n = Y n3... Y nt, X n = +2 X n3... X nt +2], [ ] [ V n = V n3... V nt, and Z n = +2 X n2... X nt +1]. However, since population means for the data are not known, in practice we require an initial within transformation to obtain small-sample analogues for each of these vectors. Ỹ n3 X n3 Ṽ n3. 1 T Ỹ n =. 1 T X n Ψ +. 1 T Ṽ n Ỹ nt +2 X nt +2 Ṽ nt +2 }{{}}{{}}{{} Ỹ n }{{} Ŷ n X n } {{ } X n Ṽ n } {{ } V n (D.1) [ ] Ẑ n = X n2... X nt +1 1 T Z n Ỹ nt = ỹ nt s nt X nt = ỹ nt 1 s nt 1 I S+1 R 1

19 Ỹ n = 1 T T +2 Ỹ nt t=3 Xn = 1 T T +2 t=3 X nt Zn = 1 T T +1 t=2 X nt for 1 T = [1,..., 1] a T 1 vector of ones, ỹ nt = 1 ln Y t, s nt = 1 ln S t and {Y nt } and {S nt } raw output and signal data. (D.1) may be rewritten Q T Ỹ n = Q T Xn Ψ + Q T Ṽ n with instruments Z n = Q T Zn for Q T = I (S+1)T (1/T )(1 T 1 T I S+1) the (S + 1)T square within-groups transformation matrix with Q T Q T = Q T (Arellano and Bover, 199). So, an equivalent representation of (21) is [ N ] 1 Ψ = Ẑ n X n N Ẑ nŷn = N Z n Q T }{{} Q T Q T X n 1 N Z n Q T }{{} Q T Q T Ỹ n (D.2) Step 1. Before we may proceed, we must establish four auxiliary steps to the main results. First, we must find ( N ) E Ẑ n V N n = E Z nq T Ṽ n = N E( Z nq T Ṽ n ) }{{} (A) (D.3) This requires breaking down component (A). Note, (A) : E( Z nq T Ṽ n ) = E( Z nṽn) 1 }{{} T E( Z n(1 T 1 T I S+1 )Ṽn) }{{} (B) (C) (B) : E( Z nṽn) = E T +2 t=3 X nt 1Ṽnt = T E( X nt 1Ṽnt) = 19

20 (C) : Z n (1 T 1 T I S+1 )Ṽn = ( T +1 t=2 X nt ) Ṽ n3 } {{ } (D3) ( T +1 t=2 X nt ) Ṽ nt +2 } {{ } (DT + 2) where, τ = 3,..., T + 2 we have, utilizing the Wold representation (C.), that (Dτ) : ( T +1 t=2 X nt ) T +1 ) Ṽ nτ = R (I S+1 M ) (Ỹnt 1 ε nτ t=2 ) + (I S+1 M 1 ) (Ỹnt 1 ε nτ 1 E(Ỹnt ε nτ ) = if τ > t M vec(σ ε ) if τ = t P t τ 1 (M 1 + P M )vec(σ ε ) if τ < t for Σ ε = E(ε nt ε nt). Thus, in expectation we have by backward induction, E(DT + 2) = E(DT + 1) = R (I S+1 M 1 )M vec(σ ε ) E(DT ) = E(DT + 1) + R [(I S+1 M )M + (I S+1 M 1 )(M 1 + P M )]vec(σ ε ) E(DT q) = E(DT q + 1) + R [(I S+1 M ) + (I S+1 M 1 )P ]P q 1 (M 1 + P M )vec(σ ε ) 1 q T 3 2

21 This implies that the expectation of (C) previous may be written E[(C)] = (T 1)R (I S+1 M 1 )M vec(σ ε ) + (T 2)R [(I S+1 M )M + (I S+1 M 1 )(M 1 + P M )]vec(σ ε ) + R [I S+1 M + (I S+1 M 1 )P ] ( T 4 i= ) i P j (M 1 + P M )vec(σ ε ) j= Because P is a triangular matrix with principal diagonal elements less than one, both P and I S+1 P are invertible. This latter point implies we have the geometric series n 1 k= P k = (I S+1 P ) 1 (I S+1 P n ). When combined with the former point this yields T 4 i= i P j = (T 3)(I S+1 P ) 1 (I S+1 P ) 1 P 1 (I S+1 P ) 1 (I S+1 P T 3 ) j= And therefore in summary of the above points which began with (D.3), ( N ) ( 3 E Ẑ n V n = N i=1 ) T i T C i 1 T C 4(I S+1 P T 3 )C (D.4) For {C i } conformable constant matrices (cf. Alvarez and Arellano (23) equation (16)). Step 2. Next, we wish to find var((t N) 1/2 N Ẑ n V n ). To do so, recall that (C.4) gives the rule of motion Y nt = P Y nt 1 + V nt for the (unobservable) variables {Y nt } of which 21

22 {Ŷnt} is a sample approximation. Note, were the actual series observable, one could write Y n( 2) = 1 T T Y nt = 1 T t=1 T (Ỹnt E(Ỹnt)) = Ỹ n( 2) E(Ỹnt) t=1 So, Y nt Y n( 2) = (Ỹnt E(Ỹnt)) (Ỹ n( 2) E(Ỹnt)) = Ỹnt Ỹ n( 2) and thus, N Ẑ n V N n = Z nq T Q T Ṽ n = R N ([ Ỹ n1... Ỹ nt ] I S+1 ) Q T Q T Ṽ n N ([ ] ) = R Y n1... Y nt I S+1 Q T V n = R N T +2 (Y nt 2 V nt ) T R t=3 } {{ } (A) N (Y n( 2) V n ) } {{ } (B) (D.) Using standard results for white noise VARMA processes (See Lutkepohl (2)), ( N ) ( ) T +2 T T +2 (A) : var (T N) 1/2 R (Y nt 2 V nt ) = R var (Y nt 2 V nt ) t=3 t=3 R Σ for Σ = Σ Y () Ω() + Σ Y (1) Ω(1) + Σ Y (1) Ω(1) Ω(j) = E(V nt V nt j), Σ Y (j) = E(Y nt Y nt j) and vec(σ Y ()) = (I (S+1) 2 P P ) 1 vec(ω) A consistent estimator for Σ is Σ = Σ Y () Ω() + Σ Y (1) Ω(1) + Σ Y (1) Ω(1) ) (D.6) 22

23 Σ Y (j) = ((T j)n) 1 N T n=2 t=1 Ŷ nt Ŷ nt j Ω(j) = ((T j)n) 1 N T +2 n=2 t=3 V nt V nt j for V nt = Ŷnt X nt Ψ. Note, this estimator is consistent because the fourth moments of the observables are finite (See Hayashi (2) pp ). Secondly, note that T (Y n( 2), V n ) d (ζ n, ξ n ) for ζ n and ξ n jointly normally distributed random variables. Thus var(t Y n( 2) V n ) var(ζ n ξ n ), or, var(y n( 2) V n ) (1/T 2 ) constant and thus in other words, var(y n( 2) V n ) is O(T 2 ). Therefore, (B) : var ( (T N) 1 T ) N (Y n( 2) V n ) = T var ( ) Y n( 2) V n = O(1/T ) ( Thus, the variance var (T N) ) 1/2 N Z nv n constant as T regardless of the rate of growth of N (cf. Alvarez and Arellano equation (1)). var ( N ) (T N) 1/2 Ẑ n V n = R Σ + O(1/T ) (D.) Step 3. Due to the previous arguments we have 1 T N N Ẑ n X n = 1 N Z T N nx n 1 N Z T 2 N n(1 T 1 T I S+1 )X n }{{}}{{} (A) (B) (D.) N Z nx n = N t=3 [ T N X nt 1X nt = R T t=3 (Y nt 2 I S+1 ) ( Y nt 1 I S+1 ) ] R 23

24 ( 1 (A) : E T N ) N Z nx n = R [(Σ Y P + M Σ ε M 1) I S+1 ] R because E(Y nt Y nt 1) = P Σ Y + M 1 Σ ε M and N Z n(1 T 1 T I S+1 )X n = [ N T +1 ( T +2 X ni i=2 j=3 X nj )] E [ T +1 ( T +2 X ni i=2 j=3 X nj )] = T E(X nt 1X nt ) + s.o.t. for s.o.t. smaller order terms in the T -degree polynomial. This implies ( 1 (B) : E T 2 N ) N Z n(1 T 1 T I S+1 )X n = O(T 1 ) and thus in summary of the points which began with (D.), [ 1 E T N ] N Z nx n R [(Σ Y P + M Σ ε M 1 ) I S+1 ] R Finally, note that (D.) also implies ( 1 var T N ) N Z nx n ( = 1 N var 1 2 T ) N Z nx n Z nx n for Z n = (1/T )1 T Z n and X n = (1/T )1 T X n since var((1/t ) N Z nx n ) = O(T 1 ) and additionally var(z nx n ) = O(T 2 ) for reasons similar to those previously discussed with respect to var(y n( 2) V n ). Therefore, in summary of those points which began with (D.) 24

25 1 we have mean-squared convergence N T N Z m.s. nx n R [(Σ Y P + M Σ ε M 1 ) I S+1 ] R, which itself always implies convergence in probability (cf. Alvarez and Arellano (23) equation (1)). 1 T N N Z nx p n R [(Σ Y P + M Σ ε M 1 ) I S+1 ] R (D.9) Step 4. Equation (D.4) implies that ( N ) ( 3 µ NT = E (T N) 1/2 Ẑ n V n = N 1/2 i=1 ) T i T C 3/2 i 1 T C 4(I 3/2 S+1 P T 3 )C Therefore, using also (D.) we have that N (T N) 1/2 Ẑ n V n µ NT = R N T +2 t=3 (Y nt 2 V nt ) R NT R NT = (T/N) 1/2 R N (Y n( 2) V n ) + µ NT Evidently lim T R NT = which is sufficient to establish that R NT is o p (1). Furthermore, due to the CLT for such processes we have R N T +2 t=3 (Y nt 2 V nt ) d N (, R ΣR) and so we have the following result analogous to Alvarez and Arellano (23) equation 2

26 (2). N (T N) 1/2 Ẑ n V n µ d NT N (, R ΣR) (D.1) Consistency. With the previous four steps completed, and in particular equations (D.4), (D.), (D.9), and (D.1) in-hand, we may now establish the main results of consistency and asymptotic normality of Ψ. (D.4) implies E((T N) 1 N Ẑ n V n ) as T while (D.) implies var((t N) 1 N Ẑ n V n ) = (T N) 1/2 (R Σ + O(1/T )). These two points mean that (T N) 1 N Ẑ n V n m.s. which implies that (T N) 1 N Ẑ n V n p. So using also (D.9) we have Ψ = Ψ + [ N ] 1 (T N) 1 Ẑ n X N n (T N) 1 Ẑ n V n } {{ } p constant. } {{ } p (D.11) or in other words, Ψ p Ψ as T regardless of the asymptotic behavior of N, which is result (22) in Proposition 2. that Asymptotic Normality. Finally, (D.9) and (D.1) jointly imply by Cramér s theorem [ N ] 1 [ N ] (T N) 1 Ẑ n X ( ) n (T N) 1/2 d Ẑ n Vn µ NT N, Avar( Ψ) where Avar( Ψ) is the explicit form of the asymptotic variance referred to in result (23) of 26

27 Proposition 2. Avar( Ψ) = E [Ẑ X ] 1 [ 1 n n RΣR E X nẑn] (D.12) A consistent estimator Âvar( Ψ) is therefore (24). The previous result also implies T N( Ψ Ψ ) [ N ] 1 (T N) 1 Ẑ n X n µ d NT N(, Avar( Ψ)) [ N ] 1 [ ( N )] 1 (T N) 1 Ẑ n X n µ NT = E (T N) 1 Ẑ n X n µ NT + RNT [ N ] 1 [ ( N )] 1 RNT = (T N) 1 Ẑ n X n E (T N) 1 Ẑ n X n µ NT Note, the previous results imply R NT is o p(1) as T regardless of the behavior of N. However, E[(T N) 1 N Ẑ n X n ] 1 µ NT = N 1/2 f(t ) for f(t ) a function of T with f(t ) as T but N 1/2 f(t ) b(t ) possibly not if N is also increasing. This b(t ) is referred to in (23). E Identification and estimation of structural parameters E.1 Identification of structural parameters when S = 1 First let us establish identification for the case of S = 1 signal. In this case, there are no nuisance parameters, so the only pertinent restrictions on θ are those imposed by Ψ and 2

28 Ω. ] [ ] π = [Ψ vech(ω) Ψ 1 Ψ 2 Ψ 3 Ω 11 Ω 21 Ω 22 = g( θ ) (E.1) Ψ i is the i-th row of Ψ and Ω ij is the (i, j) entry of Ω. Establishing the global identifiability of structural parameters in nonlinear models is generally a nontrivial task (See Rothenberg (191)). But despite the fact that the mapping g is nonlinear, it is evidently closed-form. Furthermore, there are equivalently 6 elements in both [Ψ, vech(ω) ] and θ. Thus, a sufficient condition for the global identifiability of θ is that the inverse g 1 exists. It does. β = (Ψ 3 Ψ 1 )/Ψ 2 ρ y = Ψ 3 θ = g 1 (π) : σ y = ( Ψ 2 Ψ 3 Ψ 1 ρ u = Ψ 1 Ω 21 + Ψ 2Ψ 3 1+Ψ 2 3 ( ( ) ) 2 1 Ψ σ s = Ω 1+Ψ Ψ 1 3 Ψ 2 σ 2 y σ u = Ω 11 σy 2 Ψ 2 2σs 2 Ω 22 ) /(1 + Ψ 3 1+Ψ 2 3 (Ψ 3 Ψ 1 )) (E.2) Therefore, θ is globally identifiable in any parameter space for which it is defined. The key to this conclusion is the VARMA representation (C.2). This is implied by the rule of motion (2) for ynt, without which the observation matrix C is not invertible. Hence the rationale for (2): It provides the structure necessary for primitive identifiability, while remaining entirely consistent with macroeconomic theory. 2

29 Table E.1: Exact identification of structural θ and nuisance λ parameters. λ β ρ y ρ u σ y σ u σ Total Restriction No. Elements S S S(S + 1)/2 n θ + S λ 1 S 1 Ψ S + 2 vech(ω) (S + 1)(S + 2)/2 Total n θ + S E.2 Estimation of structural parameters when S > 1 We now consider the necessarily nonlinear problem of identification and estimation of θ when S 1 given the estimators and restrictions Ψ, Ω, and λ 1 S = 1. The easiest way to begin any nonlinear identification exercise is to verify that necessary order conditions are satisfied. Specifically, in order to separately identify θ and λ, there must be at least as much information in the n θ + S vector of reduced form restrictions, π. [ ] π = λ 1 (n θ +S) 1 S Ψ vech(ω) (E.3) One way to cogently assess the dimension of either set of parameters is to enumerate them in a table. Such an analysis is presented in Table E.1. As described by this table, there are exactly as many structural and nuisance parameters as there are restrictions to be exploited for their identification. Therefore, the order condition is just satisfied for any S 1. When S > 1, π becomes too complicated to invert analytically. But Section E.3 29

30 establishes that one may compute analytically the square Jacobian, J(θ; λ) = (n θ +S) (n θ +S) π ] (E.4) [θ λ Full rank of this matrix ensures point local identifiability of the structural parameters (Rothenberg (191)). Furthermore, while π 1 is infeasible for S > 1, it is possible to compute an asymptotically efficient 2-step estimator on the basis of the Jacobian s inverse. S > 1 : 1 λ 1 S θ = θ + J( θ, λ ) 1 Ψ Ψ( θ, λ λ λ ) vech( Ω) vech(ω( θ, λ )) (E.) where θ and λ are the signal-by-signal estimators for θ and λ constructed from each individual signal s consistent estimator (2). Section E.4 pedantically details the steps to be followed in constructing this two-step estimator and establishes its asymptotic equivalence to the infeasible efficient estimator based on inverting the mapping π directly. E.3 Identification of structural parameters when S > 1 Now let us consider the case of S > 1 signals. In this case, the S nuisance parameters λ are not known and must be accounted for. It is not possible in this case to establish global identifiability, as was the case for S = 1 signal. The reason is that the mapping g becomes too complex. However, local identifiability may be established. The (n θ + S) (n θ + S) 3

31 Jacobian (E.4) may be written explicitly, 1 S J = Ψ/ θ Ψ/ λ vech(ω)/ θ vech(ω)/ λ Ψ/ θ (S+2) n θ = 1 S 1 S 1 S(S+1)/2 1 1 ψ/ θ 1 S(S+1)/2 ψ/ θ S n θ = (ρ y ρ u)λ 1 β(1) S S (ρ y ρ u)λ S β(s) 2 S S(S+1)/2 λ 1 β(1). λ S β(s) S 1 S 1 λ 1 β(1). λ S β(s) S 1 S 1 Ψ/ λ (S+2) S = 2 S ψ/ λ = ψ/ λ S S (ρ y ρ u) β(1) (ρ y ρ u) β(s) vech(ω)/ θ vec(m ) vec(m 1 ) vech(σ ε ) = + (S+1)(S+2) θ 1 + θ ε θ n 2 θ vech(ω)/ λ (S+1)(S+2) 2 S = 1 vec(m 1 ) λ = D + S+1 [(M Σ ε ) I S+1 + [I S+1 (M Σ ε )] K S+1,S+2 ] (S+1)(S+2) (S+1)(S+2) 2 31

32 1 = D + S+1 [(M 1Σ ε ) I S+1 + [I S+1 (M 1 Σ ε )] K S+1,S+2 ] (S+1)(S+2) (S+1)(S+2) 2 ε (S+1)(S+2) (S+2)(S+3) 2 2 = D + S+1 [M M + M 1 M 1 ] D S+2 vec(m ) = θ (S+1)(S+2) n θ 1 S 1 [S(S+1)/2+4] I S (S+1) 2 S vec(m 1 ) = θ (S+1)(S+2) n θ [ S 2 S+ S(S+1) 2 (2S+2) n θ ψ/ θ vec(i S ) S 2 3 ] vec(m 1 ) = λ (S+1)(S+2) S (2S+2) S ψ/ λ S 2 S vech(σ ε ) = θ (S+2)(S+3)/2 n θ 1 S (S+1) S 1 S S S S(S+1)/2 S 2σ y 1 S(S+1)/ σ u vech(σ)/ σ 32

33 By definition, for LL = Σ the lower-left decomposition of Σ we have vech(σ) = vech(ll ) σ vech(l) = D + S [L I S + (I S L)K S,S ] D S S(S+1)/2 S(S+1)/2 We say θ are locally identifiable at a point (θ, λ ) if J(θ, λ ) is full column rank. E.4 2-step estimation when S > 1. Computing an estimator for the the structural parameters when S > 1 proceeds in three steps. We now describe these steps, and provide a proof for efficiency of the 2-step estimator. 1. Obtain a consistent first stage estimator θ using signal-by-signal estimates. (a) Obtain estimates for each signal-by-signal structural parameter using (2). (b) Obtain estimates Ψ and Ω using the multiple signals under consideration jointly. (c) Individual signal estimates do not yield estimators for the off-diagonal elements of Σ. However, note that from (C.) that we have E(vntv s nt) s = σyββ 2 + (1 + ρ 2 y)σ. So a consistent estimator for Σ is Σ = (1 + ρ 2 y) [Ê(v 1 s nt vnt) s σ y 2 β ] β where β = [β(1)... β(s) ] is comprised of estimates from each signal individually and Ê(vs ntv s nt) is the lower right block of Ω. (d) θ = ( β, (vech( Σ )), ρ y, σ y, ρ u, σ u) where ( ρ y, σ y, ρ u, σ u) come from the individual signal estimates for any signal, or an average, which remains consistent. 2. λ = diag( β 1 ) ψ is the signal-by-signal estimator for λ coming from (1). ρ y ρ u 33

34 3. We are looking for the infeasible estimator [ θ λ ] with the property π = π([ θ λ ] ) for π = [1 Ψ (vech( Ω)) ] the multiple signal reduced form estimates (with restrictions λ 1 S = 1). A first order expansion of this infeasible estimator about the first stage estimators [ θ λ ] yields [ θ λ ] [ θ λ ], the proposed estimator given by (E.). The claim is that this estimator, restated here for convenience, is efficient. Note, the premise is that θ is locally identifiable, which is sufficient for J to be invertible. 1 λ 1 S θ θ = θ + J( θ, λ ) 1 Ψ Ψ(θ, λ ) λ λ λ vech( Ω) vech(ω(θ, λ )) To establish efficiency of this two step estimator, first subtract the population values [θ λ ] from both sides of the previous equation and premultiply by T N. θ T N λ θ λ = θ T N λ θ λ 1 λ 1 S + J( θ, λ ) 1 T N Ψ Ψ(θ, λ ) vech( Ω) vech(ω(θ, λ )) 34

35 We of course also have that λ 1 S Ψ(θ, λ ) vech(ω(θ, λ )) λ 1 S Ψ(θ, λ ) vech(ω(θ, λ )) + J(θ, λ ) θ λ θ λ Therefore, using the last two points, T N θ λ θ λ = T N θ λ θ λ + J( θ, λ ) 1 T N 1 Ψ vech( Ω) λ 1 S Ψ(θ, λ ) vech(ω(θ, λ )) + J(θ, λ ) θ λ θ λ 3

36 This can be reorganized as, θ T N λ θ λ (I = nθ +S J( θ, λ ) ) 1 J(θ, λ ) θ T N θ }{{} λ λ o p(1) }{{} O p(1) 1 λ 1 S + J( θ, λ ) 1 T N Ψ Ψ(θ, λ ) vech( Ω) vech(ω(θ, λ )) }{{}}{{} π([ θ λ ] ) π([θ λ ] ) }{{} T N([ θ λ ] [θ λ ] )+o p(1) (E.6) because π([ θ λ ] ) π([θ λ ] ) = J(θ, λ )([ θ λ ] [θ λ ] ) = J( θ, λ ) 1 J(θ, λ ) T N([ θ λ ] [θ λ ] ) = T N([ θ λ ] [θ λ ] ) + o p (1) Thus, (E.6) may be simplified to θ T N λ θ λ = θ T N λ θ λ + o p(1) In other words, the 2 step feasible estimator is asymptotically equivalent to the infeasible estimator. 36

37 F Bootstrap F.1 Bias Bias of the estimator is written b( θ) = E( θ θ ) with θ the population value: θ = θ +b( θ). A bootstrapped estimator for this bias is b( θ) 1 B B b=1 θ (b) θ, where θ (b) is the b-th draw from the distribution of θ and B is the number of draws. A bias-corrected estimator θ is θ = θ + b( θ) θ + 1 B B θ (b) θ = θ = 2 θ 1 B B θ (b) (F.1) b=1 b=1 Pseudocode to obtain the required B draws follows. 1. Obtain θ from the data set {{Y nt } T t=1} N. 2. Obtain residuals {{Ûnt} T t=2} N from the fitted values generated by θ for each n = 1,..., N and t = 2,..., T as follows: (a) Assume [ 1]Ûn1 = for each n. MÛn1 = 2 1 because M = [ 2 1, [m 12 m 22 ] ]. (b) Given MÛn1 = 2 1, then Ûn2 = V n2 = Y n2 P Y n1 for each n. (c) For all t = 3,..., T and each n, Ûnt = V nt M( θ s )Ûnt 1 for V nt = Y nt P Y nt Take as given {{Ûnt} T t=2} N from the last step and draw with replacement as follows: (a) Define the set of fitted vectors {Ût} T t=2 for Ût = [Û 1t... Û Nt ] from {{Ûnt} T t=2} N. (b) Draw with replacement from {Ût} T t=2 T 1 times. One must draw from joint set of fitted values across regions n because U nt may be correlated across n. Label these draws {Û (1) t } T t=2. 3

38 (c) Repeat the previous step B 1 times, resulting in the set {{Û (b) t } T t=2} B b=1. (d) The draws from the last step may now also be written {{{Û (b) nt } T t=2} N } B b=1. 4. Take as given the draws {{{Û (b) nt } T t=2} N } B b=1 from the last step and generate draws from the observables for each n = 1,..., N and b = 1,..., B as follows: (a) Set Ŷ (b) n1 = Y n1. (b) Compute Ŷ (b) n2 (c) Compute Ŷ (b) nt = = P Ŷ (b) n1 + Û (b) n2. (b) P Ŷ nt 1 + Û (b) nt + M( θ)û (b) nt 1 t = 3,..., T. (d) The resulting set of draws from the observables is written {{{Ŷ (b) nt } T t=1} N } B b=1.. For each b = 1,... B, Obtain the estimator θ from each synthetic data set {{Ŷ (b) nt } T t=1} N. This results in the set of B draws { θ s (b) } B b=1 from which to estimate the bias of θ. F.2 Confidence intervals Take as given the draws { θ (b) } B b=1 from the above estimation of bias. Recalling that θ has n θ elements, write θ = ( θ 1,..., θ nθ ) and θ (b) = ( θ (b) (b) 1,..., θ n θ ) for scalar structural parameter estimators { θ i } and b-th draw { θ (b) i }. Define the corresponding bias-corrected estimator θ i = 2 θ i 1 B B b=1 θ (b) i and bias-corrected bootstrap draws θ (b) i = 2 ( 2 θ (b) i 1 B B b=1 θ (b) i ) 1 B B b=1 θ (b) i 3

39 Note that this also implies θ (b) i θ i = ( 4 θ (b) i 3 1 B B b=1 θ (b) i ) ( 2 θ (b) i 1 B B b=1 θ (b) i ) = 2 ( θ (b) i 1 B B b=1 θ (b) i ) Then, for each i define the set S i = { θ (1) i θ i,..., θ (B) i { ( θ } i = 2 θ (1) i 1 B B b=1 θ (b) i ),..., 2 ( θ (B) i 1 B B b=1 θ (b) i )} So, for q i is the quantile function of S i, the 1 α bootstrap confidence interval of θ i is C 1 α ( θ i ) = [ θ i q i (1 α/2), θ i q i (α/2) ] The confidence interval C 1 α makes no assumptions about the distribution of U nt in order to be correctly sized. See Hansen (21) p A similar method is followed to compute intervals for φ using the draws from θ s distribution as inputs to (9). F.3 Confidence bands Take as given the draws {{{Ŷ (b) nt } T t=1} N } B b=1 and { θ (b) } B b=1 from the above estimation of bias. Label Ŷ (b) nt = [ŷ (b) nt ŝ (b) nt ], collect { β (b) } B b=1 the first elements of { θ (b) } B b=1, and compute { φ (b) } B b=1 using each θ (b) and (9). For each t = 3,..., T, n = 1,..., N, and b = 1,..., B x (b) nt = (1 φ (b) )ŷ (b) (b) (b) nt + φ β(b) 1ŝ nt 39

40 Define the deviations S and confidence interval C 1 α for x nt as in the above confidence interval computation for each n and t using x nt and { x (b) nt } B b=1. G Results Below find supplementary figures and tables to the main paper. 4

41 Figure G.1: Estimation groupings. (a) Annual sample groupings. (b) Quarterly sample groupings. Notes: Preliminary estimated NO 2 elasticity refers to the estimates β computed using NO 2 columns data in Figure A. for the annual sample, and Figure A.6 for the quarterly sample. 41

42 Table G.1: Structural parameter estimates by signal: Annual sample, Groups 1 and 2. Group 1 (N = 1). L N F E C β * (-13.,1.16) (-1.,12.63) (-24.9,-22.29) (-.1,.92) (1.4,1.) σ.* 11.9* 9.3*.23* (.9,1.9) (4.2,1.26) (.,2.1) (2.4,12.11) (-16.39,1.1) ρ y *.9* -.3 (-1.,1.33) (-11.9,12.36) (-1.4,-1.3) (.16,1.) (-2.,2.32) σ y * (-4.26,4.49) (-4.,.) (.2,2.1) (-.,6.) (-4.4,2.3) ρ u ( ) (-.9,2.2) (-1.3,2.1) (-.3,3.2) (-1.62,2.62) σ u * * (-3.2,6.1) (32.,4.4) (-.2,.19) (-1.49,.1) (.22,.4) Group 2 (N = 11). L N F E C β * * (-2.63,12.4) (-13.1,9.4) (-2.4,-2.3) (-4.34,3.2) (.39,.3) σ * 1.6*.26 (-.23,1.6) (-.3,12.2) (1.,.96) (1.96,1.99) (-3.22,3.4) ρ y * * (-3.3,1.93) (-.33,6.24) (3.49,3.) (-1.4,2.14) (-1.31,-.6) σ y 1.6.3* 13.* * (-3.42,.43) (3.24,12.) (6.34,1.41) (-.1,.44) (1.4,3.96) ρ u * (-3.,4.36) (-2.4,2.) (-.29,1.36) (-1.21,2.3) (.9,1.32) σ u * * (-3.1,6.) (.4,12.6) (-1.6,.4) (-.,2.4) (22.1,6.1) Notes: T = 16. *Significiant at 9% confidence level (confidence interval). L: Luminosity. N: NO x emissions. F: Freight volume. E: Electricity generation. C: Cement production. See Figure (a) in main paper for annual Group definitions. 42

43 Table G.2: Structural parameter estimates by signal: Annual sample, Groups 3 and Pooled. Group 3 (N = 1). L N F E C β * (-1.2,11.4) (-3.,6.) (-1.36,1.91) (-.23,.23) (.6,.44) σ.6* 1.11* 2.46*.3* 4.91* (1.3,13.44) (3.29,2.2) (12.3,3.) (1.39,13.) (1.6,6.) ρ y * (-2.,1.4) (-24.22,33.4) (-3.9,14.4) (-1.92,21.6) (-1.22,-1.16) σ y * 3.1* (-3.1,6.) (-4.93,2.64) (-.6,.99) (4.6,21.6) (-1.22,-1.16) ρ u (-3.9,4.29) (-2.39,3.4) (-.,.9) (-2.31,4.9) (-.,1.) σ u * (-.22,.2) (-.4,9.) (-6.9,13.) (-.4,9.22) (4.26,.) Pooled (N = 31). L N F E C β (-19.4,9.3) (-39.1,3.66) (-3.4,43.9) (-22.91,1.6) (-.,1.26) σ 6.93* 14.* 23.* 12.2* 1.34 (.21,11.44) (4.3,21.) (.2,34.33) (4.2,1.6) (-.4,.4) ρ y (-2.1,14.) (-23.3,26.) (-2.1,21.4) (-24.19,1.29) (-1.43,.3) σ y (-.2,4.1) (-23.2,26.) (-6.46,6.2) (-3.,.) (-.11,4.44) ρ u (-2.2,.) (-1.31,2.) (-4.1,.6) (-1.,3.) (-.92,2.14) σ u * 9.43* * (-2.24,.4) (.6,16.29) (.62,2.2) (-.69,3.34) (1.,.) Notes: T = 16. * Significiant at 9% confidence level (confidence interval). L: Luminosity. N: NO x emissions. F: Freight volume. E: Electricity generation. C: Cement production. See Figure (a) in main paper for annual Group definitions. 43

44 Figure G.2: Annual luminosity 9% confidence bands. 2 China 26 Beijing 3 Tianjin 3 Hebei Shanxi 2 12 Inner Mongolia 29 9 Liaoning 3 Jilin 26 4 Heilongjiang 2 1 Shanghai 3 Jiangsu 33 Zhejiang 33 4 Anhui 31 1 Fujian 3 Jiangxi 29 Shandong 32 Henan 29 9 Hubei 24 6 Hunan 2 Guangdong 34 6 Guangxi 31 Hainan 34 Chongqing 3 Sichuan 29 4 Guizhou 24 3 Yunnan 26 Tibet 23 6 Shaanxi 23 Gansu 2 Qinghai 2 Ningxia 26 Xinjiang Notes: Black line: Annual reported % change, regional output. Gray shading: Asian Financial Crisis period. Pink shading: confidence bands. Confidence bands are computed by bootstrap using each respective Group 1-3 estimates. China computed using Pooled estimates. 44

45 Figure G.3: 9% Confidence bands: Annual sample, NO x emissions. 2 China 26 Beijing 3 Tianjin 2 Hebei Shanxi 2 11 Inner Mongolia 2 9 Liaoning 3 Jilin 26 4 Heilongjiang 2 1 Shanghai 3 3 Jiangsu 33 Zhejiang 33 4 Anhui 31 1 Fujian 3 Jiangxi 26 Shandong 32 Henan 29 9 Hubei 24 6 Hunan 2 Guangdong 34 6 Guangxi 31 Hainan 33 Chongqing 3 Sichuan 29 4 Guizhou 22 1 Yunnan 2 Tibet 34 4 Shaanxi 23 Gansu 2 Qinghai 26 6 Ningxia 31 Xinjiang Notes: -. Black line is reported annualized percentage change in GDP. 4

46 Figure G.4: 9% Confidence bands: Annual sample, freight traffic. 3 China 2 Beijing 3 Tianjin 2 Hebei 1 Shanxi 29 9 Inner Mongolia 2 9 Liaoning 3 Jilin 26 4 Heilongjiang 2 9 Shanghai 3 Jiangsu 33 Zhejiang 33 4 Anhui 4 Fujian 3 2 Jiangxi 26 6 Shandong 32 6 Henan 29 Hubei 24 3 Hunan 2 Guangdong 34 4 Guangxi 2 Hainan 4 Chongqing 3 9 Sichuan Guizhou Yunnan Tibet 2 6 Shaanxi 2 Gansu Qinghai Ningxia Xinjiang Notes: -. Black line is reported annualized percentage change in GDP. 46

47 Figure G.: 9% Confidence bands: Annual sample, electricity generation. 29 China 26 Beijing 3 Tianjin 2 Hebei Shanxi 2 Inner Mongolia 2 Liaoning 3 Jilin 26 4 Heilongjiang 2 1 Shanghai 3 Jiangsu 33 Zhejiang 33 4 Anhui 31 1 Fujian 3 Jiangxi 26 Shandong 32 Henan 29 9 Hubei 24 Hunan 2 Guangdong 34 6 Guangxi 31 Hainan 33 Chongqing 3 9 Sichuan 29 4 Guizhou 2 Yunnan 24 Tibet 2 6 Shaanxi 32 Gansu 2 4 Qinghai 2 Ningxia 26 1 Xinjiang 2 6 Notes: -. Black line is reported annualized percentage change in GDP. 4

48 Figure G.6: 9% Confidence bands: Annual sample, cement production. 32 China 26 Beijing 3 Tianjin 9 Hebei Shanxi Inner Mongolia 2 9 Liaoning 3 6 Jilin Heilongjiang 2 1 Shanghai 3 Jiangsu 12 Zhejiang 33 4 Anhui 31 Fujian 3 94 Jiangxi 26 Shandong 32 Henan 29 9 Hubei 9 Hunan 2 4 Guangdong 14 6 Guangxi Hainan Chongqing Sichuan Guizhou 22 6 Yunnan Tibet Shaanxi Gansu 14 Qinghai Ningxia 26 Xinjiang Notes: -. Black line is reported annualized percentage change in GDP. 4

49 Table G.3: Structural parameter estimates by signal: Quarterly sample, Groups 1-4. Group 1 (N = 6). Group 2 (N = 6). N E C β (-2.31,2.2) (-12.9,29.92) (-12.22,11.9) σ.* (1.,13.2) (-1.9,22.) (-34.3,1.36) ρ y.1* (.1,1.4) (-12.,13.39) (-16.3,14.91) σ y 1.3* (.16,1.4) (-4.,6.49) (-.,6.43) ρ u (-2.36,1.) (-4.,1.12) (-.91,1.6) σ u * 9.4* (-.3,1.4) (4.6,1.4) (4.94,1.2) N E C β (-24.4,41.96) (-11.,9.61) (-4.4,43.) σ 3.9* 1.2* (1.9,4.3) (3.92,2.1) (-.19,44.4) ρ y (-32.9,24.6) (-14.9,23.3) (-.19,44.4) σ y (-6.,1.13) (-2.4,9.6) (-.,.99) ρ u (-.4,2.24) (-1.22,2.1) (-.96,2.1) σ u 1.3* 1.1*.32* (6.2,2.69) (6.3,1.92) (.4,13.26) 49 Group 3 (N = ). Group 4 (N = 6). N E C β (-4.3,6.) (-11.99,1.6) (-9.19,6.1) σ 2.49* (2.,36.1) (-1.4,24.41) (-.9,4.1) ρ y (-.9,11.64) (-.2,6.23) (-23.,31.1) σ y 2.3* (16.6,41.2) (-.9,1,) (-3.12,13.29) ρ u (-3.,.6) (-1.6,3.) (-2.91,4.93) σ u *.13* (-.39,2.1) (3.4,22.1) (1.,2.11) N E C β (-.,9.) (-.3,1.6) (-41.29,36.4) σ * (-.3,3.22) (-2.1,2.9) (6.,6.49) ρ y (-.1,12.2) (-16.61,16.34) (-1.6,19.) σ y 22.2* (11.,39.62) (-3.9,1.) (-4.64,12.) ρ u (-.6,.4) (-.4,.21) (-3.94,.4) σ u 1.34* 9.4* 9.41* (.1,23.9) (3.,21.) (3.39,1.69) Notes: T = 31. * Significiant at 9% confidence level (CI). N: NO 2 columns. E: Electricity generation. C: Cement production. See Figure (b) in main paper for annual Group definitions.

50 Table G.4: Structural parameter estimates by signal: Quarterly sample, Groups and Pooled. Group (N = 6). Pooled (N = 31). N E C β (-.43,12.43) (-4.,9.9) (-3.99,4.) σ 23.11* 13.33* 21.4 (3.22,33.) (2.,16.66) (-.24,6.9) ρ y (-16.,1.) (-1.,19.29) (-2.,2.32) σ y 11.4* 12.94* 1.4 (3.4,36.41) (.92,9.1) (-6.19,11.4) ρ u (-9.,9.26) (-.4,4.44) (-.2,2.2) σ u.26* 6.9* 9.1* (1.6,2.39) (.61,2.1) (3.,1.42) N E C β (-4.,.4) (-6.64,12.9) (-16.23,116.9) σ 23.61* 14.9* 4.29 (.6,41.1) (1.61,26.66) (-11.3,3.9) ρ y (-11.29,9.6) (-13.19,16.4) (-1.,23.23) σ y 2.* (1.61,4.33) (-2.11,12.2) (-.1,.22) ρ u (-3.,2.9) (-.1,.62) (-2.,3.) σ u *.9* (-1.11,2.62) (1.2,14.9) (3.9,1.6) Notes: T = 31. * Significiant at 9% confidence level (CI). N: NO 2 columns. E: Electricity generation. C: Cement production. See Figure (a) in main paper for annual Group definitions.

51 Figure G.: Quarterly NO 2 9% confidence bands. 3 China 64 Beijing 62 Tianjin 41 Hebei 6 4 Shanxi 36 1 Inner Mongolia 1 Liaoning 13 2 Jilin 12 Heilongjiang Shanghai 16 4 Jiangsu 6 4 Zhejiang 31 4 Anhui Fujian 4 6 Jiangxi 21 Shandong Henan 4 9 Hubei Hunan 3 Guangdong Guangxi 13 3 Hainan 43 Chongqing Sichuan 1 Guizhou 1 39 Yunnan Tibet 9 3 Shaanxi Gansu 6 41 Qinghai 6 6 Ningxia 2 4 Xinjiang Notes: 26Q1-213Q3. Black line is reported annualized percentage change in GDP. 1

52 Figure G.: 9% Confidence bands: Quarterly sample, electricity generation. 4 China 63 Beijing 1 Tianjin 2 Hebei Shanxi 32 Inner Mongolia Liaoning 1 2 Jilin Heilongjiang Shanghai 6 24 Jiangsu 6 32 Zhejiang 1 31 Anhui 2 34 Fujian 44 Jiangxi 2 Shandong Henan 4 9 Hubei 3 34 Hunan 2 Guangdong Guangxi 3 43 Hainan 3 Chongqing Sichuan 3 Guizhou 4 39 Yunnan 16 3 Tibet 4 49 Shaanxi 3 32 Gansu 6 41 Qinghai 6 66 Ningxia 4 3 Xinjiang Notes: 26Q1-213Q3. Black line is reported annualized percentage change in GDP. 2

53 Figure G.9: 9% Confidence bands: Quarterly sample, cement production. 2 China 63 Beijing 44 Tianjin 24 Hebei 39 Shanxi 32 Inner Mongolia Liaoning 3 29 Jilin 6 Heilongjiang Shanghai 6 22 Jiangsu 32 Zhejiang 1 2 Anhui 2 34 Fujian 44 Jiangxi 2 Shandong Henan 4 9 Hubei 32 Hunan 6 Guangdong Guangxi 1 43 Hainan Chongqing Sichuan 3 Guizhou 39 Yunnan 16 4 Tibet 9 36 Shaanxi 4 32 Gansu 6 46 Qinghai 6 4 Ningxia 4 3 Xinjiang Notes: 26Q1-213Q3. Black line is reported annualized percentage change in GDP. 3

54 H Time decay in signal-output relationship Above in Appendix C, we derived the representation of the model with a constant elasticity of the signal with respect to output. Alternatively, say the relationship between signal and output is now subject to a time decay as a result of technical progress, Equation (26) in the main text. This relationship replaces main paper Equation (3), while the other relationships in Equations (2)-(6) which encapsulate the model remain the same. Then dropping (i) subscripts for clarity, the ABCD representation (C.1) above for the model becomes, y nt u nt t y nt s nt t }{{} Y nt = = 1 }{{} f b(t 1) 1 } {{ } g + + ρ y ρ u 1 } {{ } A ρ y ρ u βρ y b 1 } {{ } C y nt 1 u nt 1 t 1 y nt 1 u nt 1 t } {{ } B 1 1 β 1 } {{ } D ε nt ε y nt ε u nt ε s nt }{{} ε nt (H.1) The new variable in this model is t, a time trend. So, following steps exactly similar to Appendix [ C, the observation equation implies the states are ynt 1 u nt 1 t 1] = C 1 g + C 1 Y nt C 1 Dε nt. Plugging this into the state equation and rearranging yields, y nt btψ ρ u ψ bψ y nt 1 v y nt s nt = b[1 (1 ρ y )t] + ρ y b(1 ρ y ) s nt 1 + v s nt t 1 1 t 1 }{{} } {{ } } {{ }} {{ } Y nt C[f+(I 3 A)C 1 g] CAC 1 Y nt 1 (H.2) 4

55 for the MA(1) error and parameters, v y nt ψ ψm vnt s = Dε nt + ρ y ε nt 1 = u s nt + ρ y m }{{}}{{} C(B AC 1 D) Dε nt u y nt u y nt 1 u s nt 1 (H.3) ψ = ρ y ρ u β m = σ β 2 σy 2 + σ 2 with the iid errors u y nt = εy nt + εu nt N(, σ 2 y + σ 2 u) and u s nt = βε y nt + εs nt N(, β 2 σ 2 y + σ 2 ). Note, there is no need to account for a nuisance parameter λ since we are only dealing with one signal in this instance and so λ = 1 always. First-differencing yields, y nt s nt = bψ + ρ u ψ b(1 ρ y ) ρ y y nt 1 s nt 1 + v y nt v s nt (H.4) for v y nt = uy nt uy nt 1 ψmus nt 1 + ψmus nt 2 and vs nt = u s nt u s nt 1 ρ ymu s nt 1 + ρ ymu s nt 2 each MA(2) errors. The two rows of (H.4) correspond to (2) and (2) in the main text. To test the null hypothesis H : b b /b 2 = using the first row, called Test 1 in the text, we may make use of the Wald statistic W = (T 4)N( b / b 2 ) 2 ÂÂvar ( [ b b1 b2 ] ) 1 Â d χ 2 (1) (H.) [ ] Â = 1/ b 2 b / b 2 2

56 [ ] ( N ) 1 T N T b b1 b2 = x nt 3 x nt 1 x nt 3 y nt t= t= ] x nt = [1 y nt s nt Avar ( [ b ] ) b1 b2 = E [(x nt 3 v y nt ) (x nt 3 v y nt ) ] Âvar ( [ b b1 b2 ] ) = Σ x () σ v () + 2 Σ x (1) σ 2 v(1) + 2 Σ x (2) σ 2 v(2) Σ x (j) = (N(T j)) 1 σ 2 v(j) = (N(T j)) 1 N T t=1+j N T t=1+j x nt x nt j v y nt vy nt j ] v y nt = y nt [ b b1 b2 x nt 1 To test the null hypothesis H : b b /(b 1 1) = using the second row, called Test 2 in the text, we may make use of the Wald statistic, W = (T 4)N( b /( b 1 1)) 2 ÂÂvar ( [ b b1 ] ) 1 Â d χ 2 (1) (H.6) ] Â = [1/( b 1 1) b /( b 1 1) 2 [ ] ( N ) 1 T N T b b1 = x nt 3 x nt 1 x nt 3 s nt t= t= ] x nt = [1 s nt 6

57 Avar ( [ b b1 ] ) = E [ (x nt 3 v s nt) (x nt 3 v s nt) ] Âvar ( [ b b1 ] ) = Σ x () σ 2 v() + 2 Σ x (1) σ 2 v(1) + 2 Σ x (2) σ 2 v(2) and Σ x and σ 2 v defined similar to with respect to Test 1, but using v s nt. References Abadir, K. M. and J. R. Magnus (2). Matrix algebra. Cambridge University Press. Alvarez, J. and M. Arellano (23). The time series and cross-section asymptotics of dynamic panel data estimators. Econometrica, Arellano, M. and O. Bover (199). Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 6 (1), Fernández-Villaverde, J., J. Rubio-Ramírez, T. J. Sargent, and M. W. Watson (2). ABCs (and Ds) of Understanding VARs. American Economic Review 9 (3), Hansen, B. E. (21). Econometrics. Unpublished manuscript. Hayashi, F. (2). Econometrics. Princeton University Press. Lutkepohl, H. (2). New Introduction to Multiple Time Series Analysis. Springer. Morris, S. D. (216). VARMA representation of DSGE models. Economics Letters 13. Rothenberg, T. (191). Identification in parametric models. Econometrica 39 (3), 91.