key: cord-0515301-jx2kemjy
authors: Nakano, Satoshi; Nishimura, Kazuhiko
title: The elastic origins of tail asymmetry
date: 2021-10-16
journal: nan
DOI: nan
sha: e868343f225cc43505896b390495bbe0cb8e09fd
doc_id: 515301
cord_uid: jx2kemjy

Based on a multisector general equilibrium framework, we show that the sectoral elasticity of substitution plays the key role in the evolution of asymmetric tails of macroeconomic fluctuations and the establishment of robustness against productivity shocks. Non-unitary elasticity of substitution renders a nonlinear Domar aggregation, where normal sectoral productivity shocks translate into non-normal aggregated shocks with variable expected output growth. We empirically estimate 100 sectoral elasticities of substitution, using the time-series linked input-output tables for Japan, and find that the production economy is elastic overall, relative to Cobb-Douglas with unitary elasticity. Along with the previous assessment of an inelastic production economy for the US, the contrasting tail asymmetry of the distribution of aggregated shocks between the US and Japan is explained. Moreover, robustness of an economy is assessed by the expected output growth, the level of which is led by the sectoral elasticities of substitution, under zero mean productivity shocks.

The subject of how microeconomic productivity shocks translate into aggregate macroeconomic fluctuations, in light of production networks, has been widely studied in the business cycle literature. Regarding production networks, the works of Long and Plosser (1983) , Horvath (1998 Horvath ( , 2000 , and Dupor (1999) are concerned with input-output linkages, whereas Acemoglu et al (2012 Acemoglu et al ( , 2017 ; Acemoglu and Azar (2020) base their analysis on a multisectoral general equilibrium model under a unitary elasticity of substitution, or Cobb-Douglas economy. In a Cobb-Douglas economy, Domar aggregation becomes linear with respect to sectoral productivity shocks, and because the Leontief inverse that play the essential role in their aggregation is granular (Gabaix, 2011) , some important dilation of volatility in aggregate fluctuations becomes explainable.

Moreover, peculiar aggregate fluctuations are evident from the statistical record. Figure  1 depicts the quantile-quantile (QQ) plots of the HP-detrended postwar quarterly log GDP, using the Hodrick-Prescott (HP) filter, against the standard normal, for the US (left) and Japan (right). These figures are telling that either the Cobb-Douglas or the normal shock assumption is questionable, since these assumptions together make QQ plot a straight line. Acemoglu et al (2017) explained the non-normal frequency of large economic downturns (in the US), using non-normal (heavy tailed) microeconomic productivity shocks. Baqaee and Farhi (2019) , on the other hand, claim that a non-Cobb-Douglas economy (thus, with nonlinear Domar aggregation) can lead to such non-normality in the macroeconomic fluctuations under normally distributed productivity shocks.

Indeed, the asymmetric tails of Figure 1 left seem to coincide with the case in which the aggregated shocks are evaluated in a Leontief economy. We know this because, a Cobb-Douglas economy with more alternative technologies can always find better solution (technology) than a Leontief economy with a single technology. Thus, if a Cobb-Douglas economy generates an aggregate output that corresponds to the straight line of the QQ plot, an unrobust Leontief economy that can generate less than a Cobb-Douglas economy must take the QQ plot below the straight line. This feature (of an inelastic economy) is also consistent with the analysis based on a general equilibrium model with intermediate productions with a very low (almost Leontief) elasticity of substitution conducted in Baqaee and Farhi (2019) .

If the theory that the elasticity of substitution dictates the shape of the tails of the distribution of the aggregate macroeconomic shocks were to stand, Japan would have to have an elastic economy according to the right panel of Figure 1 . Consequently, one basis of this study is to empirically evaluate the sectoral elasticity of substitution for the Japanese economy. To do so, we utilize the time-series linked input-output tables, spanning 100 sectors for 22 years , available from the JIP (2019) database. We extract factor prices (as deflators) from the linked transaction tables available in both nominal and real values. We use the sectoral series of TFP that are also included in the database to instrument for the potentially endogenous explanatory variable (price) in our panel regression analysis. Note in advance that our sectoral average elasticity estimates (σ = 1.54) exceeded unity.

To ensure that our study is compatible with the production networks across sectors, we construct a multisector general equilibrium model with the estimated sector-specific CES elasticities. We assume constant returns to scale for all production so that we can work on the system of quantity-free unit cost functions to study the potential transformation of the production networks along with the propagation of productivity shocks in terms of price. Specifically, given the sectoral productivity shocks, the fixed point solution of the system of unit cost functions allows us to identify the equilibrium production network (i.e., inputoutput linkages) by the gradient of the mapping. By eliminating all other complications that can potentially affect the linearity of the Domar aggregation, we are able to single out the role of substitution elasticity on the asymmetric tails of the aggregated shocks. The remainder of this paper proceeds as follows. We present our benchmark model of a CES economy with sector-specific elasticities and then reduce the model to Leontief and Cobb-Douglas economies in Section 2. We also refer to the viability of the equilibrium structures with respect to the aforementioned economies and show that non-Cobb-Douglas economies are not necessarily prevented from exhibiting an unviable structure. In Section 3, we present our panel regression equation and estimate sectoral elasticities of substitution with respect to the consistency of the estimator. Our main results are presented in Section 4 where we show that our nonlinear (and recursive) Domar aggregators for non-Cobb-Douglas economies qualitatively replicate the asymmetric tails presented in this section. Section 5 concludes the paper.

Below are a constant-returns-to-scale CES production function, and the corresponding CES unit cost function for the jth sector (index omitted) out of n sectors, with i = 1, · · · , n an intermediate and a single primary factor of production labelled as i = 0.

Here, σ = 1 − γ denotes the elasticity of substitution, while α i denotes the share parameter with n i=0 α i = 1. Quantities and prices are denoted by x and π, respectively. Note that the output price equals the unit cost due to the constancy of returns to scale. The Hicks-neutral productivity level of the sector is denoted by z. The duality asserts zero profit in all sectors j = 1, · · · , n, i.e., π j x j = n i=0 π i x ij . By applying Shephard's lemma to the unit cost function of the jth sector, we have:

where s ij denotes the ith factor cost share of the jth sector. For later convenience, let us calibrate the share parameter at the benchmark where price and productivity are standardized, i.e., π 0 = π 1 = · · · = π n = 1 and z 1 = · · · = z n = 1. Since we know the benchmark cost share structure from the input-output coefficients of the benchmark period a ij , the benchmark-calibrated share parameter must therefore be α ij = a ij . Taking this into account, the equilibrium price π = (π 1 , · · · , π n ) given z = (z 1 , · · · , z n ) must be the solution to the following system of n equations:

π 1 = (z 1 ) −1 a 01 (π 0 ) γ 1 + a 11 (π 1 ) γ 1 + · · · + a n1 (π n ) γ 1 1/γ 1 π 2 = (z 2 ) −1 a 02 (π 0 ) γ 2 + a 12 (π 1 ) γ 2 + · · · + a n2 (π n ) γ 2 1/γ 2 . . . π n = (z n ) −1 a 0n (π 0 ) γ n + a 1n (π 1 ) γ n + · · · + a nn (π n ) γ n 1/γ n where we can set the price of the primary factor π 0 as the numéraire. For later convenience, we write this system in a more concise form as follows:

Here, angled brackets indicate the diagonalization of a vector. Note that c : R n + → R n + is strictly concave and z ∈ R n + . Consider below a mapping E that nests the equilibrium solution (fixed point) π of (2) and maps the (exogenous) productivity z onto the equilibrium price π, i.e.,

There is no closed-form solution to (3). However, one can be found for the case of uniform elasticity, i.e., γ 1 = · · · = γ n = γ, which is as follows:

where the n row vector a 0 = (a 01 , · · · , a 0n ) is called the primary factor coefficient vector, and the n × n matrix A = a ij is called the input-output coefficient matrix. The case of Leontief economy, where 1 − γ = 0, (4), can be reduced straightforwardly as follows:

For the case of Cobb-Douglas economy, where γ = 0, we first take the log and let γ → 0 where L'Hôspital's rule is applicable since n i=0 a ij = 1 for j = 1, · · · , n.

a ij ln π i Thus, (4) can be reduced in the following manner:

It is notable that the growth of the equilibrium price dln π = (dln π 1 , · · · , dln π n ) is a linear combination of the growth of sectoral productivity dln z = (dln z 1 , · · · , dln z n ) in the case of a Cobb-Douglas economy. Otherwise, the fixed point π given z can be searched for by using the simple recurse method applied to (2). Since the unit cost function π j = (z j ) −1 c j (π; π 0 ) is monotonically increasing and strictly concave in π, we know by Krasnosel'skiȋ (1964) and Kennan (2001) that (2) is a contraction mapping that globally converges into a unique fixed point, if it exists in R n + . Note that if π 0 = 1 and (z 1 , · · · , z n ) = (1, · · · , 1), then (π 1 , · · · , π n ) = (1, · · · , 1) is an equilibrium, which must be unique. Moreover, note that obviously from (6), the existence of a positive fixed point π ∈ R n + for any given z ∈ R n + can be asserted for the case of a Cobb-Douglas economy. Specifically, it is possible to show from (6) that:

where ij denotes the ij element of the Leontief inverse [I − A] −1 . Conversely, π can have negative elements or may not even exist in R n for non-Cobb-Douglas economies. One may see this by replacing z with small (but positive) elements in (4) and (5).

From another perspective, c j (π; π 0 ) is homogeneous of degree one in (π 0 , · · · , π n ), so by Euler's homogeneous function theorem, it follows that:

Here, by Shephard's lemma, b ij denotes the equilibrium physical input-output coefficient. In matrix form, this is equivalent to:

Let us hereafter call (B, b 0 ) the equilibrium structure (of an economy). Note that if π exists in R n + , while b 0 ∈ R n + (i.e., all sectors, upon production, physically utilize the primary factor) then [I − B], where π [I − B] = π 0 b 0 , is said to satisfy the Hawkins-Simon (HS) condition (Theorem 4.D.4 Takayama, 1985; Hawkins and Simon, 1949) .

The existence of a solution y ∈ R n + for [I − B]y = d given any d ∈ R n + , and the matrix [I − B] satisfying the HS condition, are two equivalent statements (Theorem 4.D.1 Takayama, 1985) . Thus, a structure that [I − B] satisfies the HS condition is said to be viable. Conversely, for an unviable structure (that [I − B] does not satisfy the HS condition), no positive production schedule y ∈ R n + can be possible for fulfilling any positive final demand d ∈ R n + . For a Cobb-Douglas economy, we can assert that the equilibrium structure is always viable, since we know from (6) that it is always the case that π ∈ R n + . Otherwise, π may have negative elements, in which case, the equilibrium structure must be unviable. An unviable equilibrium structure may never appear during the recurse process, however, if the equilibrium price search is such that installed in the recurse process of (2); instead, the recurse will not be convergent since (2) maps into an open set R n + . Last, let us specify below the structural transformation (as the physical input-output coefficient matrix B) and network transformation (as the cost-share structure or the monetary input-output coefficient matrix S) given z, in uniform CES economy. Since an element of the gradient of the CES aggregator is:

the gradient of the uniform CES aggregator can be written as follows:

Thus, below are the transformed structure and networks, where π is given by (4):

Observe that S = A in a Cobb-Douglas economy (γ = 0) and B = A 1/z in a Leontief economy (γ = 1).

Let us start by taking the log of (1) and indexing observations by t = 1, · · · , T , while omitting the sectoral index (j). The cross-sectional dimension remains, i.e., i = 0, · · · , n.

Here, we substitute π by p to emphasize that they are observed data, and z by ζ to emphasize that they are parameters subject to estimation. For the response variable, we use the factor share a it available as the input-output coefficient.

Note that the error terms it are assumed to be iid normally distributed with mean zero. The multi-factor CES elasticity in which we are interested has been extensively studied in the Armington elasticity literature. Erkel-Rousse and Mirza (2002) and Saito (2004) apply between estimation, a typical strategy for the two-input case, to estimate the elasticity of substitution between products from different countries. Between estimation eliminates timespecific effects while saving the individual-specific effects such as the share parameter α i . For a two-factor case, the share parameter is usually subject to estimation. However, for a multi-factor case, the constraint that n i=0 α i = 1 can hardly be met. Moreover, we know in advance that α i = a i for the year that the model is standardized. Hence, we opt to apply within (FE) estimation in this study.

Below we restate (7) using time dummy variables such that γ and ζ t p t can be estimated from p it and a it via FE panel regression:

where Y it = ln a it , X it = ln p it and D k for k = 2, · · · , T denotes a dummy variable that equals 1 if k = t and 0 otherwise. For t = 1, D 2 = · · · = D T = 0 by definition, so we know that µ t = −γ ln(ζ t p t ) for t = 1, · · · , T . The estimable coefficients for (8) via FE, therefore, indicate that:

We may thus evaluate the productivity growth at t, based on t = 1, by the following formula:

We face the concern that regression (7) suffers from an endogeneity problem. The response variable, i.e., the demand for the ith factor of production by the jth sector, may well affect the price of the ith factor via the supply function. Because of such reverse causality, the explanatory variable, i.e., the price of the ith commodity, becomes correlated with the error term that corresponds to the demand shock for the ith factor of production by the jth sector.

To remedy this problem, we apply total factor productivity (TFP) to instrument prices. The JIP (2019) database provides sectoral TFP growth (in terms of Törnqvist index) as well as the aggregated macro-TFP growth, for each year interval. It is generally considered that TFP is unlikely to be correlated with the demand shock (Eslava et al, 2004; Foster et al, 2008) . In our case, the ith sector's TFP to produce the ith commodity is unlikely to be correlated with the jth sector's demand shock for the ith commodity. Hence, TFP appears to be suitable as an instrument for our explanatory variable.

On the other hand, for the primary factor i = 0, its price can be nonresponsive to sectoral demand shocks. The primary factor consists of labor and capital services, while their prices, i.e., wage and interest rates, are not purely dependent on the market mechanism but rather subject to government regulations and natural depreciations. Moreover, it is imaginable that the demand shock for the primary factor by one sector has little influence on the prices of its factors, labor and capital, if not on their quantitative ratios demanded by the sector. Thus, we apply three exogenous variables as instruments for X 0t , namely, 1) the macro TFP, 2) the macro wage rate, and 3) the macro interest rate, which are available in time series in the JIP (2019) database. More specifically, we will be examining three instrumental variables upon the FE IV regression of (8), namely, v a it , v b it , and v c it , all of which include the sectoral TFP at t, for i = 1, · · · , n, and where, v a 0t = macro TFP at t, v b 0t = macro wage rate at t, and v c 0t = macro interest rate at t. The results are summarized in Table 1 . The first column (LS FE) reports the least squares fixed effects estimation results, without instrumenting for the explanatory variable. The second column (IV FE) reports the instrumental variable fixed effects estimation results, using the IVs reported in the last column. In all cases, overidentification tests are not rejected, so we are good with the IVs we applied. Also, first-stage F values are large enough so we are satisfied with the strength of the IVs we applied. Interestingly, the estimates for the elasticity of substitutionσ = 1 −γ are larger when IVs are applied. For later study of the aggregate fluctuations, we select from the elasticity estimates based on the endogeneity test results. Specifically, we use the LS FE estimates for sectors id 6, 12, 27, 52, 62, 70, 71, 81, 88 and, hence, IV FE estimates for the rest of the sectors. Finally, we note that simple mean of the estimated (accepted) elasticity of substitution wasσ = 1.54. 1 

Let us now consider a representative household that maximizes the following CES utility:

The household determines the consumption schedule h = (h 1 , · · · , h n ) given the budget constraint W = n i=1 π i h i and price of all goods π = (π 1 , · · · , π n ). The source of the budget is the renumeration of the household's supply of the primary factor to the production sectors, so we know that W = n j=1 v j (total value added, or GDP of the economy) is the representative household's (or national) income. The indirect utility of the household can then be specified as follows:

where Π as defined as above, denotes the representative household's CES price index. Note that H = W at the baseline (z 1 , · · · , z n ) = (1, · · · , 1) where Π = 1. Thus, H is the utility (in terms of money) that the representative household can obtain from its income W given the price change π (due to the productivity shock z) while keeping the primary input's price constant at π 0 = 1. In other words, H is the real GDP if W is the nominal GDP.

From another perspective, we note that the household's income W can also be affected by the productivity shock. When there is a productivity gain in a production process, this process can either increase its output while holding all its inputs fixed or reduce the inputs while holding the output fixed. In the former case, the national income W (nominal GDP ) remains at the baseline level, which equals real GDP in the previous year H, and the GDP growth (∆ ln H = ln H − ln H = − ln Π(π)) is fully accounted for. In the latter case, however, the national income can be reduced as far as W = HΠ(π), in which case we have ∆ ln H = 0 i.e., no GDP growth will be accounted for.

Of course, the reality must be in between the two extreme cases. In this study, we conservatively evaluate national income (as nominal GDP) to the following extent:

The real GDP under the equilibrium price, which equals the household's expenditure, can then be evaluated as follows:

If we assume Cobb-Douglas utility (κ → 0) and normalize the initial real GDP (H = 1), we have the following exposition:

where µ = (µ 1 , · · · , µ n ) denotes the column vector of expenditure share parameters. The first identity indicates that the real GDP growth is the negative price index growth of the economy less the negative price index growth of a simple economy. 2 Moreover, if we assume Cobb-Douglas economy (γ j → 0) we arrive at the following:

Note that λ j is the Domar weight (Hulten, 1978) in this particular case. The parameters of the utility function are also subject to estimation. By applying Roy's identity, i.e., h i = − ∂H ∂p i / ∂H ∂W , we have the following expansion for the household's expenditure share of the ith commodity:

where, s i denotes the expenditure share of the ith commodity of the representative household. By taking the log of (10) and indexing observations by t = 1, · · · , T , we obtain the following regression equation where the parameter κ can be estimated via FE.

As per normal, the error term δ it is assumed to be iid normally distributed with mean zero.

Here, we replace π by p to emphasize that they are observed data. For the response variable, we use the expenditure share m it of the final demand available from the input-output tables. The cross-sectional dimension of the data for regression equation (11) is i = 1, · · · , n, whereas it is i = 0, 1, · · · , n for (7). Thus, we apply sectoral TFP available for t = 1, · · · , T from the JIP (2019) database as instruments to fix the endogeneity of the explanatory variable. The estimation result for κ using time dummy variables as in (8) (such that we may retrieve the estimates for Π t ) is presented in Table 1 (id = 101). 

For a quantitative illustration, we study the distribution of aggregate output ln H when sectoral shocks ln z are drawn from a normal distribution. Specifically, we use 10,000 ln z j ∼ N (0, 0.2) iid samples for j = 1, · · · , n, where the standard deviation (i.e., annual volatility of 20%) is chosen in reference to the annual volatility of the estimated sectoral productivity growth ln ζ j (see Appendix 1). Let us first examine the granularity of our baseline production networks (i.e., 2011 input-output linkages). Below are both Cobb-Douglas price indices in terms of productivity shocks ln z for the Cobb-Douglas and simple economies:

Here, we set the share parameter µ at the standard expenditure share of the final demand m = (m 1 , · · · , m n ) for the year 2011. Both indices must follow normal distributions because they are both linear with respect to the normal shocks ln z. The variances differ, however, and the left panel of Figure 2 depicts the difference. Observe the dilation of the variance in the Cobb-Douglas economy where the power-law granularity of the Leontief inverse causes the difference (Gabaix, 2011; Acemoglu et al, 2012) .

Replacing the equilibrium price of (9) with the output of (3) yields the following Domar aggregator, where exogenous productivity shocks (ln z) are aggregated into the growth of output (ln H) for a CES economy.

ln H = − ln Π E (z; π 0 = 1) + ln Π 1/z Note that this aggregator involves recurse in E, as specified by (2), for a CES economy with non-uniform substitution elasticities. In this section, Cobb-Douglas utility is assumed for comparison with previous research, whence the Domar aggregator becomes: Again, we set the share parameter µ at the standard expenditure share of the final demand.

In case of Leontief economy, closed form is available from (5) as follows:

The case for Cobb-Douglas economy is also obtainable from (6) as follows:

As is obvious from the linearity of (14) the aggregate fluctuations must be normally distributed in the Cobb-Douglas economy. The resulting QQ plot is depicted in the right panel of Figure 2 . Note further that E[ln H] = 0 because of the linearity of (14) and E[ln z] = (0, · · · , 0); and this is recognizable in the same figure. That is, the expected economic growth is zero against zero-mean turbulences. In other words, the unit elasticity of the Cobb-Douglas economy precisely absorbs the turbulences as if there were none to maintain zero expected growth. Such robustness is absent in a Leontief economy with zero elasticity of substitution. The left panel of Figure 3 illustrates the resulting QQ plot of the aggregate fluctuations generated by the same zero-mean normal shocks by way of (13). 3 In this case, we observe negative expected output growth, i.e., ln H = −1.57%, whose absolute value, in turn, can be interpreted as the robustness of the unit elasticity of substitution. 4 Moreover, we observe that normal shocks to the Leontief economy result in aggregate fluctuations with tail asymmetry similar to that depicted in the left panel of Figure 1 .

In the CES economy with a sector-specific elasticity of substitution, we use the Domar aggregator of general type (12) with recurse. The right panel of Figure 3 depicts the resulting QQ plot of the aggregate fluctuations generated by zero-mean normal shocks. In this case we observe positive expected output growth i.e., ln H = 1.10%. The value demonstrates the robustness of the elastic CES economy relative to Cobb-Douglas. We also observe that normal shocks to the elastic CES economy result in aggregate fluctuations with tail asymmetry similar to that depicted in the right panel of Figure 1 . For sake of credibility, we show in Figure 4 (right) the empirical aggregate output fluctuations focusing over the period 1994-2015 from which our sectoral elasticities are estimated. It is obvious that extreme observations belong to periods around the GFC (global financial crisis), which was a massive external (non-sectoral) shock to the Japanese economy. The plot otherwise seems rather skewed positively, as predicted by our empirical result (σ = 1.54).

Figure 4 (left) shows the aggregate output fluctuations focusing over the period 1997-2020 from which we estimated sectoral elasticities for the US (see Appendix 2 for details). In this case also the extreme observations belong to periods around the GFC and Covid19 pandemic. The plot, however, seems to be in a straight line, which is consistent with our empirical result (σ = 1.08). Our elasticity estimates for the US economy also coincide with the estimates of elasticity of substitution across intermediate inputs ( M all = 1.05) by Miranda-Pinto and Young (Forthcoming) based on 1997-2007 input-output accounts. These elasticity estimates, however, differ from those obtained by Atalay (2017) based on 1997-2013 input-output accounts, that Baqaee and Farhi (2019) employed in their simulation (ε = 0.001). An inelastic economy as such is rather consistent with negatively skewed aggregate fluctuations spanning the postwar US economy as depicted in Figure 1 (left) than those of recent times depicted in Figure 4 (left).

It is well documented that the Japanese have been more creative in discovering how to produce than in what to produce. The empirical results obtained in this study provide some evidence to believe that such a spirit is engraved in the nation's economy. Undeniably, the technologies embodied in a production function have been acquired over the long course of research and development. Japan must have developed its elastic economy through the grinding process of discovering more efficient and inexpensive ways to produce while overcoming the many external turbulences it confronted. Whatever the cause may be, an elastic economy equipped with many substitutable technologies must be favorable with respect to robustness against turbulence. Ultimately, human creativity expands the production function in two dimensions: productivity and substitutability, and the elasticity of substitution in particular, that acts upon the robustness of the economy, must be worthy of further investigation.

A geometric Brownian motion (GBM) can be specified by the following stochastic differential equation (SDE):

where, µ denotes the drift parameter, σ denotes the volatility, and B t ∼ N (0, t). Ito's Lemma implies that the above SDE is equivalent to the following:

where this SDE is solvable by integration. The solution follows below: ln X T = ln X 0 + (µ − σ 2 /2)T + σB T Since B T ∼ N (0, T ), the first and the second moments for ln(X T /X 0 ) can be evaluated as follows:

There are several ways of estimating the volatility and the drift parameters of a GBM, empirically from size of historical data (X 1 , · · · , X ). The obvious one is the following, which is based on the sample moments:

Alternatively, Hurn et al (2003) devised parameter estimates of the following based on the simulated maximum likelihood method.

The two methods produce very similar results for our data. 

Notes: The id number corresponds to the numerical position of an industry of the input-output table of 71 industries BEA (2022). Values in parentheses indicate

Wald) F statistic for 2SLS FE estimation. The rule of thumb to reject the hypothesis that the explanatory variable is only weakly correlated with the instrument is for this

*2 Overidentification test by Sargan statistic. Rejection of the null indicates that the instruments are correlated with the residuals

Endogeneity test by Davidson-MacKinnon F statistic. Rejection of the null indicates that the instrumental variables fixed effects estimator should be employed

3, respectively indicate v a , v b , v c , and l, f, d respectively indicate first lag, first forward

Funding and Conflict of interest JSPS Kakenhi Grant numbers: 19H04380

Endogenous production networks

The network origins of aggregate fluctuations

Microeconomic origins of macroeconomic tail risks

How important are sectoral shocks?

The macroeconomic impact of microeconomic shocks: Beyond hulten's theorem

Cabinet Office (2021) Quarterly estimates of gdp -release archive

Aggregation and irrelevance in multi-sector models

Import price elasticities: Reconsidering the evidence

The effects of structural reforms on productivity and profitability enhancing reallocation: evidence from colombia

Reallocation, firm turnover, and efficiency: Selection on productivity or profitability?

FRED (2021) Federal Reserve Economic Data

The granular origins of aggregate fluctuations

Note: Some conditions of macroeconomic stability

Cyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks

Sectoral shocks and aggregate fluctuations

Growth Accounting with Intermediate Inputs

On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential equations*

Japan industrial productivity database

The simple mean of all annual volatilities is 0.251. The normality of TFP growth rates is examined by Shapiro-Wilk W test, where rejection of normality is indicated by the label 'no', and a blank is left otherwise. *1 Electronic data processing machines, digital and analog computer equipment and accessories *2 Image information, sound information and character information production

This section is devoted to our estimation of sectoral substitution elasticities for the US, in the same manner as we did for Japan. First, we shall create n × n input-output tables using Make and Use Tables of n = 71 industries in nominal terms for 24 years , available at BEA (2022). Next, we create tables in real terms by using price indices available as Chain-Type Price Indexes for Gross Output by Industry. Note that real value added of an industry is estimated by Double Deflation, so that price indices for value added can be derived from nominal and real value added accounts. As for instruments, we utilize the Integrated Multifactor Productivity (MFP), taken from 1987-2019 Production account capital table (BLS, 2022) of the BEA-BLS Integrated Industry-level Production Accounts (KLEMS), for n factor inputs. 5 For instrumenting primary factor prices, we apply three different instruments, namely, Total factor productivity (i.e., aggregate TFP), Capital price deflator, and Labor price deflator, obtainable from the Annual total factor productivity and related measure for major sectors (BLS, 2022) . Thus, our instrumental variables are v a (sectoral MFP with aggregate TFP), v b (sectoral MFP with Capital price deflator), and v c (sectoral MFP with Labor price deflator), all of which are of n + 1 dimension.Estimation of sectoral elasticities of substitution was conducted according to the estimation framework presented in section 3. The results are summarized in Table 3 . The first column (LS FE) reports the least squares fixed effects estimation results, without instrumenting for the explanatory variable. The second column (IV FE) reports the instrumental variable fixed effects estimation results, using the IVs reported in the last column. In all cases, overidentification tests are not rejected. Also, first-stage F values are large enough. The estimates for the elasticity of substitutionσ are larger when IVs are applied. According to the endogeneity test results, we accept the LS FE estimates for sectors id 6, 9, 10, 11, 21, 34, 37, 41, 48, 55, 56, 62, 63, 64, 65, 71 , in stead of the IV FE estimates. A simple mean of the estimated (accepted) elasticities isσ = 1.08.