The Granular Origins of Aggregate Fluctuations : Supplementary Material

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1 The Granular Orgns of Aggregate Fluctuatons : Supplementary Materal Xaver Gabax October 12, 2010 Ths onlne appendx ( presents some addtonal emprcal robustness checks ( descrbes some econometrc complements (on the attenuaton bas n the R 2 of the granular regressons, and on why demeanng by ndustry-year averages can be better than demeanng by year averages ( descrbes the tme-seres of the Granular Resdual avalable onlne. 5. Robustness Checks for Granular Resdual Regressons I consder possble mcro-level mprovements n the constructon of the granular resdual. Perhaps the loadng on the aggregate factors could depend on sze. To explore that possblty, I consder models such as: g t = ag I t + b ( ln S,t 1 ln S j,t 1;,j I gi t + c ( ln S,t 1 ln S j,t 1;,j I 2 gi t + ε t (36 In ths model, the loadng of frm can depend not only on the ndustry average g I t, but also on the sze of the frm. To make coeff cents easer to nterpret, I recenter by ln S j,t 1, the average log sze, so that the mean of the second term s 0. Table VI reports the frst stage of a varety of specfcatons (wth K = Q = 100. Sze controls are only sometmes sgnfcant, but ther ncremental explanatory power s very small compared to the earler specfcaton that smply controls for g t and g I t. The second stage s reported n Table VII. The mpact of the sze control s very small. Hence, for practcal purposes, one may recommend the two smplest versons of the granular resdual, (33 and (34. Stll, one lesson from Table VI s that large frms have f anythng a smaller loadng on the common shocks than small frms (column 5. Hence, constructng the granular resdual by removng a constant mean slghtly overcorrects large frms, and, prma face, works aganst fndng a hgh explanatory power of the granular resdual. Ths bas aganst the granular resdual s, however, small, as Table VII shows. Tables VIII and IX show some more regressons. In partcular, I use K = 100 and Q = Ths s, t evaluates the expected behavor of the top 100 frms by examnng the behavor of the top 1000 frms. Note that a dff culty s that n the early part of the sample there are less than 1000 frms; then I take the maxmum number of frms. The results are smlar to Tables VI and VII, except that the R 2 of the frst stage (explanng frm-level growth rate and the second stage (explan GDP growth are a bt lower. That may be 41

2 (1 (2 (3 (4 (5 (6 (Intercept 7.2e e e e (0.002 ( (0.002 ( ( ( ḡ t 1** 1** 0.9** (0.076 (0.076 (0.093 (0.068 (0.081 ḡ I(,t, 1** 1** 0.97** (0.02 (0.021 (0.022 (ln S,t 1 ln S t ( ( ( ( (ln S,t 1 ln S t 1 ḡ t * (0.12 (0.18 (0.11 (0.16 (ln S,t 1 ln S t ( ( (ln S,t 1 ln S t 1 2 ḡ t 0.26* 0.17 (0.13 (0.12 (ln S,t 1 ln S I(,t ( ( (ln S,t 1 ln S I(,t 1 ḡ I(,t -0.31** -0.4** (0.05 (0.057 (ln S,t 1 ln S I(,t ( (ln S,t 1 ln S I(,t 1 2 ḡ I(,t 0.16** (0.05 N R Adj. R Table VI: Explanng Frm-Level Productvty Growth. Ths table presents the results of the dfferent regressons of productvty growth g t for the top 100 frms as defned by the prevous year s sales on mean productvty growth measures at the global and ndustry level. The column numbers correspond to dfferent specfcatons for the regressors. They are used n the next table. Standard errors n parentheses. 42

3 (1 (2 (3 (4 (5 (6 (Intercept 0.017** 0.017** 0.018** 0.019** 0.02** 0.019** ( ( ( ( ( ( Γ t 2.5** 4.5** 2.8** 2.6** 4.2** 4.4** (0.69 (0.82 (0.68 (0.7 (0.83 (0.84 Γ t 1 2.9** 4.3** 2.7** 2.9** 3.8** 4.6** (0.67 (0.78 (0.68 (0.68 (0.81 (0.77 Γ t 2 2.1** 2.7** 2.1** 2.1** 2.9** 2.9** (0.71 (0.79 (0.7 (0.72 (0.82 (0.81 N R Adj. R Table VII: Explanatory Power of the Granular Resdual, under alternatve specfcatons. Ths table reports the regressons of per capta GDP growth on the granular resdual plus two lags, usng the varous specfcatons of Table VI to extract the dosyncratc frm-level shocks. Standard errors n parentheses. due to the fact that the sample s less homogenous, as we try to control the behavor of large frms by the behavor of qute smaller frms. 6. Econometrc Complements 6.1. Attenuaton Bas n the Granular Resdual I analyze the propertes of the granular resdual. The concluson s that t suffers from attenuaton bas, but the bas goes to 0 as the number of frms K becomes large. On the other hand, takng a large K (or Q ntroduces new dff cultes the homogenety assumpton (37 s lkely to be a less good approxmaton, as demonstrated n the prevous secton. I consder frst a one-factor model (no ndustry shocks. For frm : g t = X t + ε t (37 where X t s a common shock, and ε t s an dosyncratc shock. The granular resdual s: Γ K = K =1 S (g g Y (38 whle the econometrcan would lke to know the deal granular resdual a weghted mean of 43

4 (1 (2 (3 (4 (5 (6 (Intercept 3.2e e e e (0.001 ( (0.001 ( ( ( ḡ t 1** 1** 1** (0.037 (0.037 (0.045 (0.038 (0.045 ḡ I(,t, 1** 1** 1** (0.012 (0.013 (0.015 (ln S,t 1 ln S t ( (0.001 (0.002 (0.002 (ln S,t 1 ln S t 1 ḡ t ** 0.16** (0.033 (0.037 (0.034 (0.037 (ln S,t 1 ln S t ** ( ( (ln S,t 1 ln S t 1 2 ḡ t (0.02 (0.021 (ln S,t 1 ln S I(,t (0.002 (0.002 (ln S,t 1 ln S I(,t 1 ḡ I(,t -0.22** -0.22** (0.014 (0.015 (ln S,t 1 ln S I(,t 1 2 7e-04 ( (ln S,t 1 ln S I(,t 1 2 ḡ I(,t -2.4e-05 ( N R Adj. R Table VIII: Ths table presents the results of the dfferent regressons of productvty growth g t for the top 1000 frms as defned by the prevous years sales on mean productvty growth measures at the global and ndustry level. The column numbers represent correspond to dfferent specfcatons for the regressors. They are used n the next Table. Standard errors n parentheses. 44

5 (1 (2 (3 (4 (5 (6 (Intercept 0.018** 0.018** 0.016** 0.023** 0.016** 0.023** ( ( (0.003 ( ( ( Γ t * 1.3* 1.3* 1.3* 1.4* (0.54 (0.53 (0.57 (0.58 (0.57 (0.58 Γ t 1 1.9** 2.8** 1.9** 1.7** 2.3** 2.1** (0.56 (0.54 (0.6 (0.61 (0.6 (0.6 Γ t (0.56 (0.53 (0.59 (0.6 (0.59 (0.6 N R Adj. R Table IX: Resdual Productvty Growth Baselne Regressons, Q=1000, K=100. Ths table reports the results of regressons of the per capta GDP growth on the granular resdual plus two lags where the granular resdual s computed wth Q = 1000 and K = 100 whle usng estmates of the resdual productvty growth from the prevous Table. Standard errors n parentheses. the dosyncratc shocks of the top K frms. Γ K = K =1 S ε Y (39 (the specfc choce of the denomnator does not matter here, as I nvestgate the R 2 s, and R 2 s do not change when one multples some varables by a constant. GDP growth follows, as n the model of the NBER WP verson of ths paper: y t = φγ Kt + u t (40 where u t s a dsturbance orthogonal to (ε t =1...K. One would lke to know how much R 2 of the dosyncratc shocks of the top K frms explan,.e., the R 2 of the deal granular resdual: RΓ 2 = cov (y t, Γ Kt 2 K var (y t var ( Γ (41 Kt The emprcal analyss only gves the R 2 of the granular resdual Γ : R 2 Γ K = cov (y t, Γ Kt 2 var (y t var (Γ Kt (42 Econometrcally, the stuaton s trcky, because economcally, X t s correlated wth Γ K. 45

6 A quantty of nterest s the squared herfndahl of the top K frms: H K = K =1 S2 ( K =1 S 2 (43 Lemma 1 The R 2 of the granular resdual s a downward based estmate of the R 2 of the deal granular resdual, by a factor 1 1 KH K : R 2 Γ K = R 2 Γ K ( 1 1 KH K Proof. By rescalng, t s enough to analyze the case where σ ε = 1. I call K =1 S = s, and X = K 1 K =1 X for a varable X. Then: Γ = K =1 (S /s 1/K ε, whch gves, droppng the K subscrpts when there s no ambguty: Γ t = Γ t + ε t wth cov (Γ t, ε t = 0 whch means that Γ s a nosy proxy for Γ. Also cov (Γ t, Γ t = varγ t = ( H 1, varγ t = H K and RΓ 2 = cov (y, Γ t 2 vary var (Γ t = φ2 cov (Γ t, Γ t 2 vary var (Γ t = φ2 (varγ 2 ( 1 1 vary var (Γ t ( = RΓ ( H HK H 1 = RΓ HK K HK 2 = cov (y, Γ t 2 ( vary varγ varγ HK varγ ( Emprcally, for the K = 100, frms, 1 1 KH K = 2/3. Hence f emprcally the RΓ 2 K = 1/3, the R 2 of the deal granular resdual s RΓ 2 = 1/2. Ths bas s an attenuaton bas, as the granular K resdual s a nosy proxy for the deal granular resdual. If the dstrbuton s very concentrated, then H K 1/K. Formally, the proof of Proposton 2 shows that f the Pareto exponent of the dstrbuton s 1 ζ < 2, KH K K 2 2/ζ, so lm K (KH K 1 = 0, and as K, R 2 Y,Γ K /R 2 Y,Γ K 1. Ths s the sense n whch, for large K, the granular resdual dentfes the explanatory power of the deal granular resdual. The same reasonng apples, wth messer expressons, wth ndustry specfc shocks, model: g t = x t + x I + ε t. The R 2 of the ndustry-demeaned Γ t s a downward estmate R 2 of the deal granular resdual Γ,nd. The bas goes to 0 as the number of frms becomes large. 46

7 6.2. Why controllng for ndustry-year averages g I t mght be better than smply controllng for the year average g t The paper controls for g t and g I t. It justfes ths by sayng The term g t g I t may be closer to the deal ε t than g t g t, as g I t may control better than g t for ndustry-wde dsturbances, e.g., ndustry-wde real prce movements. In case ths s useful, here s some more elaboraton on ths argument. Indeed, take a producton functon Y = F (A 1 L 1,..., A n L n wth one factor, labor, and F farly general but homogenous of degree 1. Then, the sales are S = p A L, hence the percentage changes (callng X = dx/x are Ŝ = p +  + L, so the paper s construct g s: g Ŝ L =  + p Thus, g s ndeed TFP growth Â, plus a cost dsturbance p, whch econometrcally s akn to some measurement nose (except that, mportantly, t can be correlated wth Â. If the p are very correlated wthn an ndustry (that wll be the case wth a Dxt-Stgltz structure nsde the ndustry, p p I, then g I t p I p (f there are many frms wth uncorrelated shocks n the ndustry, and g g I t wll be closer to  than g g t. Ths general pont beng made, here s a clear example n whch the demeanng by the ndustry g I t s unambguously better theoretcally than the demeanng by g t. Consder an economy wth ( m sectors where sector I produces: Q I = j I (A jl j 1/φ φ m for ψ > 1, and GDP s Y = Q α I wth m =1 α I. Then, the prce of good Q I satsfes p I Q I = α I Y, so p I + Q I = Ŷ ( 1 Also, (by optmzaton n the producton of Q I the prce of good I s p = p Q φ 1 I Q I, so the sales of frm are: ( 1 Q φ S = p Q = p I Q I Q I so: Ŝ = 1 φ Ŝ = 1 φ ( Q Q I + p I + Q I ( Q Q I + Ŷ I=1 I Note also that Q I = I S Q I S To smplfy further, assume that labor cannot be reallocated n the short term (the more general 47

8 case s n the NBER WP of ths paper, so that L = 0 and Q = Â. Then, the measure growth rate g Ŝ L satsfes: g = 1 φ ( Q Q I + Ŷ Next, consder the g term wth many sectors and many frms (but a few bg frms n a few bg sectors whle the ndustry average s g = Ŷ g I = Q I φ Denote by s = S /Y the output share. The g-demeaned granular resdual (eq. 33 n the paper s: + Ŷ Γ g s (g g = 1 s ( Q φ Q I ( = 1 s  1 s Q I φ φ I I Γ g = 0 (44 The granular term, purgng by g, s just Γ g 0. However, the ndustry-demeaned granular term (eq. 34 n the paper s: Γ nd where Γ = s  s the deal granular resdual. The key results are: ( ( 1 s g g I = s φ Q = 1 s  = 1 φ φ Γ. (45 Γ g = 0, Γ nd = 1 φ Γ (46 Hence, a fnte sample, Γ g wll be pure nose, whereas Γ nd wll contan nformaton on the deal granular resdual Γ. Thus, to proxy for the deal granular resdual Γ, the emprcal granular resdual Γ nd = K S,t 1 ( =1 Y gt t 1 g I t wll be better suted than the emprcal granular resdual Γ g = K S,t 1 =1 Y t 1 (g t g t. Ths example s of course not general (I just consdered a tractable lmt case, but t llustrates the pont that controllng for g It may control better for ndustry-wde dsturbances, such as ndustrywde real prce changes. 7. Descrpton of the Granular Resdual Tme-Seres The onlne data contans the followng seres. 48

9 Seres 1 ( GR, ndustry-demeaned, K = 100 s the granular resdual Γ t seres, as used n the paper (K = Q = 100, usng the ndustry-level demeanng. Seres 2 ( GR, K = 100 s the granular resdual Γ t seres, as used n the paper (K = Q = 100, usng the g t demeanng. Seres 3 ( Narratve GR s the narratve granular resdual that corresponds solely to the frm-year events selected n the narratve. I defne the Narratve Granular Resdual Γ N t as beng 0 f no event s selected, and equal to Γ N ( t = S t 1 gt g I t /Yt 1 f a frm s selected for that year (or I sum over the frms f several frms are selected. Seres 4 ( GR, K = 5 s the granular resdual based on the top 5 frms only, Γ t (K = 5, Q = 100, usng the ndustry-level demeanng. 49