Interindustry linkages of prices—Analysis of Japan’s deflation

The interactions among macroprices with leads and lags play a significant role in explaining the behavior of an aggregate price index. Thus, to understand inflation and deflation, it is essential to explore the mechanism according to which these macroprices interact with each other. On the basis of a new method, we show that, irrespective of the sources of shocks, a robust flow of changes occurs in domestic prices from upstream to downstream. Moreover, we demonstrate that macroprices change in clusters, and we identify these clusters. Firms are not symmetric. Overall, our analysis suggests that the inertia arising from input/output linkages in production explains the behavior of aggregate prices.

Appendix A Validity Check of the CHPCA Here we address how effective is the CHPCA in extracting the comovement behavior of prices with lead-lag relations.The validation analysis of the method is carried out using synthetic multivariate time series into which definite lead-lag relations and noises are incorporated.
We begin with preparing the following time series data with the same numbers of N and T that the price data have: where i = 1, 2, • • • , N and t = 0, 1, • • • , T .The first term in the right-hand side of Eq. (S1) represents a sinusoidal signal of period P with a given phase δ i , which is randomly chosen in the range (−π/2, π/2).The second term plays a role of random noises disturbing the signal; determines the relative strength of noises as compared with the signal and z i (t) is generated randomly according to the standardized normal distribution.We then apply the CHPCA to the test time series by taking difference for each of them: Figure S1 exemplifies the time series synthesized at three values of together with their difference; we set m = 5 and hence P = 76.6 throughout this analysis.The difference of the test data is almost indistinguishable from that of completely random noises at = 0.3.This is because the intensity of the signal is two orders of magnitude smaller than that of noises.The signal-to-noise ratio S/N in ∆x i (t) is approximately given by S/N = P where P S and P N are the signal power and the noise power, respectively.This formula is derived by noting that the difference of x i (t) can be replaced by its derivative in the condition of P 2π: Taking ensemble average of Eq. (S4) over noises, we obtain The time average of Eq. (S5) finally leads to the power of the total fluctuations: where the first and second terms in the right-hand side of Eq. (S6) give P S and P N , respectively.The time-averaged power as given in Eq. (S6) should be relevant in the CHPCA, because it entirely depends on static (equal time) correlations between complexified time series.
In Fig. S2, we show the probability distribution functions of the eigenvalues of the CHPCA applied to the synthesized data for three values of in the range 0.2 ≤ ≤ 0.3.The largest eigenvalue λ 1 at = 0.2 is clearly isolated from the bulk of the eigenvalues, and it is statistically significant according to the same criterion adopted in the main text; it is larger than the upper limit λ c of the largest eigenvalue predicted for the corresponding data which are completely random.On the other hand, λ 1 at = 0.3 is merged into the bulk of the eigenvalues and it is not statistically significant with λ 1 < λ c .At = 0.25, λ 1 ( λ c ) is in a marginal situation.
Figure S3 depicts the first eigenvectors on complex plane obtained at the same values of as in Fig. S2.Coinciding with the statistical significance of the largest eigenvalue, the eigenvector at = 0.2 reproduces well the lead/lag relations embedded in the test data while one cannot detect the hidden relations out of the eigenvector for = 0.3.This result is more clearly visible in Fig. S4, where the phases extracted from the first eigenvector of the CHPCA are compared with the corresponding phases given on synthesizing the data at = 0.2, 0.25, and 0.3.We note that the CHPCA works for the test data with the S/N just less than 5%.
From the above test on the CHPCA we can infer that the method is quite effective in extracting the comovement behavior of prices with lead-lag relations.And also we confirm that the lead/lag relations among time series data detected from the eigenvectors of the CHPCA are reliable once their corresponding eigenvalues are judged to be statistically significant.
< l a t e x i t s h a 1 _ b a s e 6 4 = " B y e v l J p T 6 P 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " B y e v l J p T 6 P 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " B y e v l J p T 6 P 6 : Probability distribution functions of the eigenvalues of the CHPCA applied to the synthesized data for three values of .The vertical arrow in each panel shows the upper limit of the largest eigenvalue determined at 3σ confidence level for the corresponding data which are completely random.< l a t e x i t s h a 1 _ b a s e 6 4 = " 9 n X P b s P u 5 a H : Eigenvectors depicted on complex plane, each of which is associated with the largest eigenvalue in the corresponding panel of Fig. S2.
+ / e h / e 5 W C 1 4 + c 0 p W I L 3 9 Q u N 0 p e w < / l a t e x i t > ✏ = 0.25 < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 n X P b s P u 5 a H Figure S4: Comparison of given phases of the synthesized data and the corresponding phases extracted from the first eigenvectors of the CHPCA in Fig. S3.
the same mathematical relation between V + and V − as long as the condition σ 1 σ 2 is satisfied.The random time series is fixed to the comovement of domestic prices at any phase angle.The remaining issue to be addressed is thereby whether the fixed phase difference θ between the two collective coordinates in the first eigenmode is statistically significant or not.To test statistical significance of the phase angle between the comovement of domestic prices and the exchange rate, we reiterated the CHPCA calculation for the data set in which the exchange rate and import prices are substituted by a random time series with the variance kept the same; the new results serve as a null model.The strength of coupling between the two collective coordinates is represented by the magnitude of σ 12 and hence by difference of the two dominant eigenvalues as shown in Eq. (S8). Figure S5 demonstrates distribution of λ 1 − λ 2 in the null model.On the other hand, the actual result for λ 1 − λ 2 is 2.401 and its p-value is given as 0.006 according to the null hypothesis.This comparison allows us to infer that the fixed phase angle between the comovement of domestic prices and the exchange rate is statistically meaningful.
In conclusion, the correlation structures in the two dominant eigenmodes are fully understandable with a two-variable model.And also we confirm that the exchange rate is certainly a driving factor for the first eigenmode.Eigenvector of the largest eigenvalue obtained for the former half of the whole period without the IPI prices, depicted in the complex plane.
t e x i t s h a 1 _ b a s e 6 4 = " B 0 O D W y 2 / T m NW Q N F o 0 A K 5 q L R p A H 4 = " > A A A C C H i c b V D L S s N A F J 3 U V 6 2 P R l 2 6 G S x C B S l J F X R Z 0 I X L C v Y B b Q i T 6 a Q d O n k w c y O W k B / w F 9 z q 3 p 2 4 9 S / c + i V O 2 y x s 6 4 E L h 3 P u 5 V y O F w u u w L K + j c L a + s b m V n G 7 t L O 7 t 1 8 2 D w 7 b K k o k Z S 0 a i U h 2 P a K Y 4 C F r A Q f B u r F k J P A E 6 3 j j m 6 n f e W R S 8 S h 8 g E n M n I A M Q + 5 z S k B L r l n u 3 z I B B D + 5 K c + q c O a a F a t m z Y B X i Z 2 T C s r R d M 2 f / i C i S cB C o I I o 1 b O t G J y U S O B U s K z U T x S L C R 2 T I e t p G p K A K S e d P Z 7 h U 6 0 M s B 9 J P S H g m f r 3 I i W B U p P A 0 5 s B g Z F a 9 q e 1 P x P 6 + T m P D G T 5 m M E 0 M l m S 8 K E 4 5 M h K b P o z 5 T l B g + t g Q T x e y t i A y x w s T Y i B a 2 B G J S s K F 4 y x G s k m a 1 4 l 1 W q v d X p d p F F k 8 e T u A U y u D B N d T g D u r Q A A I c X u A V 3 p x n 5 9 3 5 c D 7 n r T k n m z m G B T h f v x g R l l s = < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = " l u O m 7 f x k i g n l G R S N r j U A C h 6 l l L w = " > A A A B + H i c b V A 9 S w N B E J 2 L X z F + R S 1 t D o N g I e E u C l o G b C w T M B + Q H G F v s 5 c s 2 d 0 7 d u e E G P I L b L W 3 E 1 v / j a 2 / x E 1 y h U l 8 M P B 4 b 4 a Z e W E i u E H P + 3 Z y G 5 t b 2 z v 5 3 c L e / s H h Figure S1: Examples of the synthesized data at three values of and their difference.The signal-to-noise ratio S/N is 4.2, 0.042, 0.019 for = 0.02, 0.2, 0.3, respectively.
e h / e W C + c p W I L Q u N p e w < / l a t e x i t > ✏ = 0.25

Figure S5 :
FigureS5: Distribution of the eigenvalue separation, λ 1 − λ 2 .The CH-PCA with a random time series in place of the exchange rate and import prices (sampled 10,000 times), where the variance of random time series is kept the same as σ 2 = 7.42.

Figure S6 :
Figure S6: Collective behavior of individual prices in the former half.Eigenvector of the largest eigenvalue obtained for the former half of the whole period without the IPI prices, depicted in the complex plane.

Figure S7 :
FigureS7: Same as Fig.S6, but in the latter half.Eigenvector of the largest eigenvalue obtained for the latter half of the whole period without the IPI prices, depicted in the complex plane.

Figure S10 :
Figure S10: Price clusters in a super-critical condition.The panels (a) and (b) shows the 20 largest clusters as detected by the percolation analysis with g c = 0.3, corresponding to the panels (a) and (b) in Fig. 10. .

Figure S11 :
Figure S11: Price clusters in a sub-critical condition.Same as Fig. S10, but with g c = 1.0.