ONLINE APPENDIX: INFORMATION, MISALLOCATION AND AGGREGATE PRODUCTIVITY Joel M. David Hugo A. Hopenhayn Venky Venkateswaran
Appendix I: Detailed Derivations I.A
Case 1: Both factors chosen under imperfect information
As we show in equation (5) in the text, the firm’s capital choice problem can be written as ✓
max Kit
N Kt
◆↵ 2
✓
1
↵ Yt ✓ Eit [Ait ] Kit
1+
↵2 ↵1
◆
Rt Kit
and optimality requires
↵
✓
↵1 ↵1 + ↵2
✓ ↵
)
◆✓
↵1 ↵1 + ↵2
◆✓
N Kt N Kt
◆↵ 2
1
↵ Yt ✓ Kit
◆↵ 2
1
Eit [Ait ]
Eit [Ait ] Yt Rt 1 ✓
=
Rt
=
Kit
1 1
↵
Capital market clearing then implies Z )
"✓
◆ ✓1 # 1 1 ↵ Z 1 ↵1 Yt Kit di = ↵ (Eit [Ait ]) 1 ↵ di = Kt ↵ 1 + ↵ 2 Rt "✓ ◆ ✓ ◆ ✓1 # 1 1 ↵ ↵2 N ↵1 Yt Kt ↵ =R 1 Kt ↵ 1 + ↵ 2 Rt (Eit [Ait ]) 1 ↵ di N Kt
◆↵ 2
✓
from which we can solve for
Kit = R
(Eit [Ait ]) 1 (Eit [Ait ]) 1
1 ↵ 1 ↵
di
Kt
From here, it is straightforward to express firm revenue as
1 ✓
0
Pit Yit = Kt↵1 N ↵2 Yt Ait @ R
(Eit [Ait ])
1
(Eit [Ait ]) 1
1
↵
1 ↵
di
1↵ A
and noting that aggregate revenue must equal aggregate output, we have
Yt =
Z
Pit Yit di =
or in logs, yt =
1 yt + ↵1 kt + ↵2 n + log ✓
Z
1
Kt↵1 N ↵2 Yt ✓
R ↵ Ait (Eit [Ait ]) 1 ↵ di ⇣R ⌘↵ 1 (Eit [Ait ]) 1 ↵ di
Ait (Eit [Ait ]) 1
1
↵ ↵
di
↵ log
Z
(Eit [Ait ]) 1
1 ↵
di
Now, note that under conditional log-normality, ait |Iit ⇠ N (Eit [ait ] , V)
✓ ◆ 1 Eit [Ait ] = exp Eit ait + V 2
)
The true fundamental ait and its conditional expectation Eit ait are also jointly normal, i.e. 2 6 4
ait Eit ait
3
02
3 2
2 a
¯ 7 6 7 B6 a 5 ⇠ N @4 5,4 a ¯
2 a
V
2 a
2 a
31
V7 C 5A V
We then have that
log
Z
Ait (Eit [Ait ])
1
↵ ↵
di
◆ 1 ↵ log exp ait + Eit ait + V di 1 ↵ 21 ↵ ✓ ◆2 1 1 2 1 ↵ 2 a ¯+ + V a 1 ↵ 2 a 2 1 ↵ ↵ 1 ↵ 2 + V + V a 1 ↵ 21 ↵ 1 1 2 1 2 ↵ 1 ↵ 2 a ¯+ + ↵ V + V 1 ↵ 2 a 2 (1 ↵)2 a 21 ↵
= =
=
and log
Z
(Eit [Ait ]) 1
1 ↵
✓
Z
di =
1 1
↵
a ¯+
1 2
↵
✓
1 1
↵
◆2
2 a
V +
1 1 V 21 ↵
Combining these,
log
Z
Ait (Eit [Ait ]) 1
↵ ↵
di
↵ log
Z
(Eit [Ait ]) 1
1 ↵
di = a ¯+
1 21
2 a
1 ↵ V 21 ↵
↵
Substituting and rearranging, we obtain the expressions in (10) and (11) in the text,
yt
= =
1 1 a2 1 ↵ yt + ↵ 1 k t + ↵ 2 n + a ¯+ V ✓ 21 ↵ 21 ↵ ✓ ◆ 2 ✓ 1 ✓ 1 a ↵ ˆ 1 kt + ↵ ˆ2n + a ¯+ ✓V ✓ 1 2 ✓ 1 1 ↵ 2 | {z } a⇤
=
↵ ˆ 1 kt + ↵ ˆ2n + a
with aggregate productivity is given by a = a⇤
1 2 ✓V.
It now remains to endogenize Kt . The rental rate in steady state satisfies R =
2
1
1 + . Then, from the
optimality and market clearing conditions, we have ↵1 N R = ↵2 K W
)
K/W
i.e., the aggregate capital stock is proportional to the wage. To characterize wages, we return to the firm’s profit maximization problem
1
↵1 ↵2 Yt ✓ Eit [Ait ] Kit Nit
max
Kit ,Nit
W Nit
RKit
which, after maximizing over capital, can be written as
max
(1
Nit
↵1 )
⇣ ↵ ⌘ 1 ↵↵1 ⇣ 1
1
R
1
Yt ✓ Eit (Ait ) Nit↵2
⌘1
1 ↵1
W Nit
Optimality and labor market clearing imply ⇣↵ ⌘1 2
W
⇣↵ ⌘1 2
⇣↵ ⌘1
1 ↵1 ↵1 ↵ 2
1 ↵1 ↵1 ↵2
W
1
⇣↵ ⌘1 1
↵1 ↵ 1 ↵2
R
Z ⇣
↵1 ↵ 1 ↵2
R
⇣
1
Yt ✓ Eit [Ait ]
1
Yt ✓ Eit [Ait ]
⌘1
⌘1
1 ↵1
1 ↵1
↵2
↵2
di
=
Nit
=
N
As before, letting ↵ = ↵1 + ↵2 , we see that
W
/ =
=
✓Z 2 4
Eit [Ait ] 1 Z ✓ ✓
1 ↵
di
◆ 11
↵ ↵1
1
✓
1 exp Eit ait + V 2
1 exp a ¯+ 2
✓
2 a
1
or in logs, w/
↵V ↵
✓
1 ↵1
Yt ✓ 1
◆◆ 1 1 ↵
◆◆
Yt
1 1
↵1
1 ✓
◆
di 1
!1
↵ 1 ✓
1 ↵1
Yt 5
1 ↵1
a ¯+
31
1 1
1 ↵1 2
✓
=
2 a
1
↵V ↵
✓
1 exp a ¯+ 2
◆
+
✓
2 a
1
1 1 yt ✓ 1 ↵1
Recalling that K/W
)
dk dw = dV dV
which, in conjunction with (10) and (11), implies dy =↵ ˆ1 dV
✓
dk dV
◆
1 ↵ ˆ1 ✓= 2 1 ↵1
3
1 ↵ 1 dy + 2 1 ↵ ✓ dV
1 ✓ 2
V ↵
◆
◆ 1 1 + V Yt ✓ 2
1
1 ↵1
and finally, collecting terms and rearranging, and using the fact that ↵ b1 = dy = dV
I.B
✓ ✓ 1 ↵1 ,
we obtain
1 1 da 1 ✓ = 2 1 ↵ ˆ1 dV 1 ↵ ˆ1
Case 2: Only capital chosen under imperfect information
The firm’s labor choice problem can be written as 1
↵1 ↵2 Yt ✓ Ait Kit Nit
max Nit
and optimality requires Nit =
⇣↵
2
W
Wt Nit
1
↵1 Yt ✓ Ait Kit
⌘1
1 ↵2
Labor market clearing then implies Z
Nit di
) where we let A˜it = Ait1
Wt
= = =
1 ↵2
Z ⇣ ⌘ ↵2 ✓1 ↵1 1 Yt Ait Kit W
=
Nit
=
↵1 1 ↵2 .
and ↵ ˜=
1 ✓
R
↵1 1 (Ait Kit )
↵1 (Ait Kit )
1 ↵2
1 1 ↵2
di
1 ↵2
di = N
N=R
↵ ˜ A˜it Kit N ↵ ˜ di A˜it Kit
This then implies
! ↵2 1 ↵ ˜ A˜it Kit N R ↵ ˜ di A˜it Kit ✓Z ◆1 ↵ 2 ✓ ◆↵ 2 ↵1 1 ↵1 ↵ ˜ A˜it Kit di Ait Kit Ait1 ↵2 Kit1 ↵2
↵1 Yt Ait Kit ↵2
↵2 N1
↵2 N1
1
Y✓ ↵2 t Y ↵2 t
1 ✓
✓Z
↵ ˜ A˜it Kit di
◆1
1
↵2
From here, it is straightforward to express the firm’s capital choice problem as in (8):
max
(1
Kit
↵2 )
✓
↵2 Wt
◆ 1 ↵↵2
2
1
Yt ✓ 1
1 ↵2
h i ↵ ˜ Eit A˜it Kit
Rt Kit
Optimality requires
Rt )
Kit
✓
↵2 Wt
=
(1
↵2 )
=
↵2 ) ↵ ˜ R
(1
◆ 1 ↵↵2 1
1
↵ ˜
✓
2
1
Yt ✓ 1
↵2 Wt
1 ↵2
◆ 1 ↵↵2 4
h i ↵ ˜ Eit A˜it ↵ ˜ Kit 1
2 1
↵ ˜
Yt ✓(1
1 ↵2 )(1
1
↵) ˜
⇣
h i⌘ 1 1 ↵˜ Eit A˜it
Capital market clearing then implies Z
Kit di
=
)
Kit
=
1 ✓ ◆ ↵2 1 ↵2 )˜ ↵ 1 ↵˜ ↵2 1 ↵2 1 ↵˜ ✓(1 Yt R Wt ⇣ h i⌘ 1 1 ↵˜ Eit A˜it ⇣ h i⌘ 1 1 ↵˜ Kt R Eit A˜it di
(1
1 ↵2 )(1
↵) ˜
Z ⇣
h i⌘ 1 1 ↵˜ Eit A˜it di = Kt
and we can rewrite the labor choice as ⇣ h i⌘ ↵˜ ˜it Eit A˜it 1 ↵˜ A ↵ ˜ ˜ Ait Kit Nit = R N= ⇣ h i⌘ 1 ↵˜↵˜ N ↵ ˜ di R A˜it Kit A˜it Eit A˜it di Combining the solutions for capital and labor, we can express firm revenue as
Pit Yit
=
=
=
9 ↵1 8 h i⌘ 1 1 ↵˜ > = > < Eit A˜it Yt Ait K t 1 ⇣ h i⌘ > > >R 1 ↵ ˜ : R E A˜ di ; : it it 1 ✓
8 > <
⇣
⇣ h i⌘ ↵11+↵↵˜2 ↵˜ ↵2 ˜ Ait Ait Eit A˜it 1 Yt ✓ Kt↵1 N ↵2 ⇢ ⇣ h i⌘ 1 ↵1 ⇢ ⇣ h i⌘ 1 ↵˜↵˜ R R 1 ↵ ˜ ˜ Eit Ait di A˜it Eit A˜it di ⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it Yt Kt↵1 N ↵2 ⇢ ⇣ h i⌘ 1 ↵1 ⇢ ⇣ h i⌘ 1 ↵˜↵˜ R R 1 ↵ ˜ Eit A˜it di A˜it Eit A˜it di 1 ✓
and again using the fact that yt = log
yt =
9 ↵2 ⇣ h i⌘ 1 ↵˜↵˜ > = A˜it Eit A˜it N ⇣ h i⌘ 1 ↵˜↵˜ > A˜it Eit A˜it di ;
1 yt + ↵ 1 k t + ↵ 2 n ✓
R
↵1 log
↵2
↵2
Pit Yit di, we can write Z ⇣
h i⌘ 1 1 ↵˜ Eit A˜it di + (1
↵2 ) log
Z
⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di
Again, we exploit log-normality to obtain
log
Z
⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di
= =
◆ 1 ↵ ˜ ˜ V di 1 ↵ ˜ 21 ↵ ˜ 1 1 2 1 2 ↵ ˜ ⇣ 2 ˜⌘ 1 ↵ ˜ ˜ a ˜+ ˜ V + V a ˜ + ↵ a ˜ 2 1 ↵ ˜ 2 2 (1 ↵ 21 ↵ ˜ ˜)
log
Z
✓ exp a ˜it +
5
↵ ˜
Eit a ˜it +
and similarly, Z ⇣
log
Eit
h
A˜it
i⌘ 1 1 ↵˜
di
Z ✓
◆◆ 1 1 ↵˜ 1˜ log exp Eit a ˜it + V di 2 ˜ 1 1 a˜2 V 1 1 ˜ a ˜+ V 2 + 21 1 ↵ ˜ 2 (1 ↵ ↵ ˜ ˜)
= = ait 1 ↵2 :
Combining, and using the fact that a ˜it = Z ⇣
Z
⇣ h i⌘ 1 ↵˜↵˜ di + (1 ↵2 ) log A˜it Eit A˜it di 1 ↵2 ↵1 (1 ↵2 ) 2 1 (1 ↵2 )(2˜ ↵ ↵ ˜2) ⇣ 2 ˜ ⌘ a ˜+ V a ˜ + a ˜ 1 ↵ ˜ 2 2 (1 ↵ ˜ )2 1 ↵ a ¯ (1 ↵2 ) 2 1 ↵1 ⇣ 2 ˜ ⌘ + V a ˜ + 1 ↵ ˜ 1 ↵2 2 21 ↵ ˜ a˜ 1 1 1 ↵1 2 ˜ a ¯+ (1 ↵2 )V a 21 ↵ 21 ↵ ↵1 log
= = =
Eit
h
✓
A˜it
i⌘ 1 1 ↵˜
Substituting and collecting terms, we obtain expressions (10) and (12) in the text,
yt
= =
↵ b 1 kt + ↵ b2 n +
✓ ✓ |
1
1 2
a ¯+
✓
✓
✓ {z
1
◆
2 a
1
↵ }
a⇤
↵ b 1 kt + ↵ b2 n + a
1 2
with aggregate productivity given by a = a⇤
1 ˜ (✓b ↵1 + ↵ b2 ) ↵ b1 V 2
˜ (✓b ↵1 + ↵ b2 ) ↵ b1 V.
To endogenize Kt , we begin by characterizing the steady state wage W in terms of 1
and Y ✓ :
W
= = =
↵2 N1
Y ↵2
↵2 N1
↵2
↵2 N1
Y
Y ↵2
1 ✓
1 ✓
1 ✓
✓Z
(Z
(1
↵ ˜ A˜it Kit di
A˜it
✓
(1
↵2 )˜ ↵ R
◆1
⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di
↵2
↵2 )˜ ↵ ⇣ ↵2 ⌘ 1 R W
↵1 1 ↵ ˜
R
⇣ ↵ ⌘ 1↵2 ↵↵˜˜ 2
W
6
↵2 ↵2
Y
Y
1 ↵ ˜ ✓ 1 ↵ ˜
1 ✓(1
↵2 )
⇢Z
Eit
h
A˜it
i◆ 1 ↵˜↵˜
di
)1
⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di
↵2
1 ↵2
and rearranging,
W
= / = =
⇣
⌘ 11
↵2
↵ ˜ ↵1
(1
↵2 )˜ ↵ R
N 1 ↵2 ⇢Z ⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di ( ⇢
exp
✓
1 1
1 a ˜+ ↵ ˜ 2
1 exp a ¯+ 2 (1
1
↵1 ↵1
1 ↵ 1 ↵1
↵2 ↵ ˜ 1 ↵1
↵2 Y
↵ ˜)
2 a
⇢Z
⇣ h i⌘ 1 ↵˜↵˜ A˜it Eit A˜it di
(1
↵2 )(1 1 ↵1
↵) ˜
1 1 ✓ 1 ↵1
1 2˜ ↵ ↵ ˜2 ⇣ 2 a ˜ + 2 (1 ↵ ˜ )2
1 ↵2 )(1
Y
1 1 ✓ 1 ↵1
2 a ˜
1↵ ˜ (1 2 1
⌘
˜ ˜ ˜ +1 ↵ V V 21 ↵ ˜ ↵2 ) ˜ V Y ↵ ˜
1 ✓
1
◆
1 ↵
Y
1 ✓
)1
1 ↵1
1 ↵1
or in logs,
w/
✓
1 1
↵1
◆
a ¯+
1 1 1 2 (1 ↵1 ) (1 ↵)
2 a
1 ↵ ˜ (1 ↵2 ) ˜ 1 1 V+ yt 2 (1 ↵1 )(1 ↵ ˜) ✓ 1 ↵1
As before, K/W
)
dk dw = = ˜ ˜ dV dV
1 ↵ ˜ (1 ↵2 ) 1 1 dy + ˜ 2 (1 ↵1 )(1 ↵ ˜ ) ✓ 1 ↵ 1 dV
and substituting into the derivative of aggregate output, dy ˜ dV
= =
✓
dk ˜ dV
◆
1 (✓b ↵1 + ↵ b2 ) ↵ b1 2 1 ↵ ˜ (1 ↵2 ) 1 1 dy ↵ ˆ1 + ˜ 2 (1 ↵1 )(1 ↵ ˜ ) ✓ 1 ↵ 1 dV ↵ ˆ1
1 (✓b ↵1 + ↵ b2 ) ↵ b1 2
Finally, collecting terms and rearranging, and using the facts that 1 ↵ = (1 ↵ b1 , we obtain
I.C
dy = ˜ dV
The stock market
1 da 1 (✓b ↵1 + ↵ b2 ) ↵ b1 = ˜ 2 1 ↵ b1 dV
✓
1 1
↵ ˆ1
↵ ˜ ) (1
↵2 ) and
◆
⇣
✓ ✓ 1
⌘
↵1 =
Rewriting the marginal investor’s indi↵erence condition in recursive form yields a fixed-point characterization of the price function: P (a (30)
1 , a, z)
=
Z
⇡ (a 1 , a, a + v z) dH (a|a 1 , a + v z, a + v z) Z Z + P (a, a0 , z 0 ) dG (a0 , z 0 |a) dH (a|a 1 , a +
v z, a
+
v z)
Next, we connect expected profits ⇡ (·) in the price function to the firm’s problem. For brevity, we only
7
show the derivation for case 1 (case 2 is very similar). The firm’s profit is
= =
=
=
✓ ✓ ✓ |
N Kt
◆↵ 2
N Kt
◆↵ 2
Yt Ait
N Kt
◆↵ 2
Yt ✓
1 Ait
✓
1
↵ Yt ✓ Ait Kit 1 ✓
R
(Eit [Ait ])
◆
↵2 ↵1
(Eit [Ait ]) 1
↵ 1
(Eit [Ait ]) {z
1
di
↵
di !↵
1
1
↵ ↵
2
Kt
↵
Kt 1
RKit
1
(Eit [Ait ]) 1
R
1
1+
(Eit [Ait ])
!↵
✓
↵2 1+ ↵1
Ait (Eit [Ait ]) 1
}
◆
↵ ↵
R
(Eit [Ait ]) 1 ↵ di ⇣ ⌘ 2 1+ ↵ ↵1 RKt
(Eit [Ait ]) | {z
↵
↵ 1
R
1
1
1
(Eit [Ait ]) 1
1
1
↵
2
RKt
di }
(Eit [Ait ]) 1
1 ↵
From the definition of the firm’s information set, Eit [Ait ] = E [Ait |ait
1 , ait
+ eit , ait +
v zit ]
=
V 2 µ
⇢ (ait
1
a ¯)
The profit function ⇡ (·) is obtained after integrating out the noise term in the firm’s signal
⇡ (ait
1 , ait , ait
+
v zit )
=
1
✓ ◆ ↵ eit Ait (E [Ait |ait 1 , ait + eit , ait + v zit ]) 1 ↵ d e ✓ ◆ Z 1 e it ((E [Ait |ait 1 , ait + eit , ait + v zit ])) 1 ↵ d 2
Z
e
Given ⇡ (·), we solve the functional equation (30) using an iterative procedure with a discrete grid of shock realizations. We then verify that the conjectured threshold and invertibility properties of the equilibrium hold for all points on the grid.
I.D
Other Distortions
Here, we show how a general class of output and capital distortions, (1
⌧Y i ) and (1 + ⌧Ki ), can be
mapped into a composite distortion on capital. We start with Case 1, where the firm’s distorted objective is given by, as in Hsieh and Klenow (2009),
max
Kit ,Nit
Eit [(1
↵1 ↵2 ⌧Y it ) Ait Kit Nit
W Nit
(1 + ⌧Kit ) RKit ]
As before, we use (4) to substitute out for Nit and derive the following FOC for Kit
Kit =
⇢
1 Eit
(1 ⌧Y it ) Ait (1 + ⌧Kit ) 8
1 1
↵
where
1
is a constant. Letting
⌧it ⌘ log
(1 ⌧Y it ) = log (1 (1 + ⌧Kit )
⌧Y it )
log (1 + ⌧Kit )
yields equation (19) in the text. Directly, the covariance matrix of (ait , ⌧it ) can be derived using 2 6 4
3
2 a
2
61 5⌘4 2 0 ⌧
a⌧ 7
a⌧
3
2
61 07 6 5 ⌃ 6 60 4 1 0
0 1
3
07 7 17 7 5 1
Finally, we can express ⌧it = ait + "it where
⌘
2 a⌧ 2 a
and "it ⇠ N 0,
2 ⌧
2 2 a
mrpkit = yit
. The marginal revenue product of capital is given by
kit = ait + (↵
1) kit = ait
(1 + ) Eit ait
"it
with a cross-sectional dispersion equal to 2 mrpk
=
2 a
=
V+
+ (1 + )
Analogously, for Case 2, Kit =
2
(
2
2 a
2 a
V
2 Eit
1
⌧Y it ) 1 ↵2 A˜it (1 + ⌧Kit )
(1
2 a
V +
2 "
2 "
V +
"
2 (1 + )
#) 1 1 ↵˜
which yields the following expression for the composite distortion:
⌧˜it ⌘ log
"
(1
1
⌧Y it ) 1 ↵2 (1 + ⌧Kit )
#
=
log (1 ⌧Y it ) 1 ↵2
log (1 + ⌧Kit )
To characterize the stochastic properties of ⌧˜it , we follow the same steps as in Case 1.
9
I.E
Identification
Transitory shocks.
A log-linear approximation of prices (around the deterministic, undistorted case):1
P exp (pit )
=
˜ it AK ↵ exp (ait + ↵kit ) E
) P + P pit
⇡
AK ↵
P pit
⇡ =
pit
=
˜ it RK exp (kit ) + E ˜ it Pit+1 + Const. E
˜ it (ait + ↵kit ) RK E ˜ it kit + E ˜ it P pit+1 + Const. RK + P + AK ↵ E ˜ ˜ it ait + (↵AK ↵ RK) Eit [Eit ait ] + E ˜ it P pit+1 + Const. AK ↵ E 1 ↵ ˜ it ait + E ˜ it P pit+1 + Const. AK ↵ E AK ↵ ˜ ˜ it pit+1 + Const. Eit ait + E P
˜ denotes the marginal investor’s expectation. We guess where E ˜ it ait + Const. pit = ⇠ E Substituting, ˜ it ait + Const. = ⇠E
AK ↵ ˜ ˜ it ait+1 + Const. Eit ait + ⇠ E P
)
⇠=
Y 1 = P 1
↵
Now, recall that both of the signals in the marginal investor’s information set are equal to uit + Then, ˜ it [ait ] = E ˜ it [µit ] = E where =
1
+
+
1
2 v
1
2 µ
2 v
(µit +
1
2 v
+
Combining, and setting = ⇠ , yields (22).
1. The aggregate constant multiplying revenues is normalized to 1.
10
2 z
1
2 v
2 z
v zit )
v zit .
Variances (of growth rates): 2 p
⌘
2 k
Var (pit
⌘
Var (kit
=
2
⇣
1
pit kit 1
+
2 e
2 v
1
=
2
1
2 µ
2 a
⌘
2 v
2 z
+
1
2 z
=2
1)
h
⌘2
⇣
1
+ !
1
2 v
ait
1)
⇣
+
2 µ
2 z
2 µ
⇠
2 u
2 e
+
2 2
= 2(
1
+
2 e
Var (ait
1)
=2
+
1 1
2 e
1
+
2 e
2)
⌘2
2 µ
2 2 v z
+
1
2 µ
⇣
2 e 2 v
2
+ ⌘2 2 µ
2 2 2 v z
✓
✓
+
2 2 2 2 v z
1
2 v
2 2 v z
✓
⌘2
2 z
✓
V
⇠
2 2 1 e
+
2 z
=2
2 2
=2
1 1
↵
2 2 v z 2 µ
1+
i✓
1 1
↵
◆
2 u
1 1
↵
◆2
◆2
◆2
2 µ
Covariances (of growth rates):
Cov ( p,
k)
⌘
2⇠
1+ 0⇣
2)
1
2⇠ @
= Cov ( p,
a)
=
2⇠
1
+
2 e
2 µ
2 v
+ ⌘
2 z
1
2 µ
+
2 µ
+
1
+
2 e
⇣
1
2 v
1
2 v
1 1 2 z
2 z
↵
⌘
◆
2 2 v z
2 µ
1 A
✓
1 1
↵
◆
2 µ
= 2⇠
Correlations:
⇢pa
⇢pk
=
Cov ( p,
a)
p a
=
Cov ( p,
k)
p k
=
r⇣
=q
µ
2⇠2
4
=q
1 ⌘⇣ 2 2 1 + v2 z 1
2 µ
2⇠ 2 µ
+
2⇠ 2⇠2
4 V
2 µ
2 µ
+
2 2 v z
=q
2 µ
2 µ 2 2 v z
(
2 u
1 1+
2 v
2 µ
2 z
V)
⌘
The volatility of returns:
2 p
0
= ⇠2 @
2 z 2 z
+1
+1+
2 v
2 µ
2 z
12 A
✓
1+
2 2 v z 2 µ
11
◆
2 u
= ⇠2
2 z
+1 2+ 1 z ⇢2
pa
!2
1 ⇢2pa
2 µ
✓
1 1
↵
◆
Permanent shocks.
We start with the price function, (suppressing the time subscript and normalizing
the GE term multiplying revenues to 1), P (a
˜ (ea K ↵ =E
1 , a, z)
˜ (a, a0 , z 0 ) RK) + EP
˜ denotes the marginal investor’s expectation. For small information frictions, profits are where, as before, E approximately ea K ↵
RK ⇡ e 1
a ↵
Substituting, P (a
1 , a, z)
˜ 1 a↵ + EP ˜ (a, a0 , z 0 ) ⇡ Ee
Guess: P (a
1 , a, z)
ˆ Ee ˜ 1 a↵ = P·
To verify the guess ˆ Ee ˜ 1 a↵ P·
⇡ =
⇣ 0 ⌘ ˜ 1 a↵ + P· ˆE ˜ E ˜ 0 e 1a ↵ = Ee ˜ 1 a↵ + P· ˆE ˜ (u0 , z 0 ) · Ee ˜ 1 a↵ Ee ⇣ ⌘ ˆE ˜ (u0 , z 0 ) Ee ˜ 1 a↵ 1 + P·
˜ 0 denotes the expectation of marginal investor in the following period, which in turn is a function where E of the future shock realizations (u0 , z 0 ). Since (u0 , z 0 ) are iid, our guess is verified. Then,
log P (a
1 , a, z)
⇡ Const. +
1 1
↵
˜ Ea
or equivalently, pit
1
⇡
1 1
↵
˜ it E
1 ait 1
+ const.
Next, when ⇢ = 1, firm and investor beliefs are given by Eit ait = ait
1
˜ it ait = ait E
+ Eit µit
12
1
˜ it µit , +E
which leads to
(1
where the coefficients
ait
=
pit
= =
(1
↵)
kit
,
pit
=
↵) kit
=
and
1
1 1
↵
ait
1
ait
+
+
1 1
1 1
(uit +
↵
(µit + eit ) +
2
v zit )
(µit +
+ Const
v zit )
+ Const.
are the same as in the i.i.d. case. In growth rates,
2
µit 1 1
↵ 1
1
↵
=
(ait
=
(1
(ait (1
) µit ait
1 1
ait
1
2)
2)
+
+
1
+(
1
2 ) µit 1
1 1 1
1
+(
2 ) (µit 1
+
+
✓
v zit
µit + ⇠
↵
+
(µit +
↵
µit
2 ) µit
+
µit
(zit
v 1)
+
1
(eit
zit
1
v zit 1 )
1 1)
(eit
eit
eit
1)
1)
+
+
2 v
(zit
2 v
(zit zit
zit
1)
1)
Second moments (of growth rates): 2 p
(31) (32) (33) (34)
= =
2 k
= = =
✓ ✓ ✓ ✓ ✓
1 1
↵ 1
1
↵ 1
1
↵ 1
1
↵ 1
1
↵
◆2
(1
◆2
◆2 h
2 µ
(1
◆2 ◆2
)
2 µ
2
1
2 µ
2 +2 2)
1
2
2 µ
2
1
◆2
2 2 µ
1
↵
2
(1 +
2 2 v z ) 2 µ
2 µ
1
+(
V +2
2 µ
+
+2
2)
2 µ
4V + 2
✓
1
◆2
1
↵
+2
2 2 1 e
V2 2 µ
+2
+2
V2 2 e
2 2 2 v z
2 2 2 2 v z
+2
V2
i
2 2 v z
2 µ
Thus, the variance of investment is invariant to uncertainty! This is because, with persistence, undoing the e↵ects of past forecast errors also contributes to investment volatility. When shocks are permanent, this exactly o↵sets the contemporaneous dampening that occurs due to imperfect information. Thus, with permanent shocks, uncertainty shifts the timing of the response of capital to fundamentals, but not the
13
overall magnitude. Now,
Cov( p,
a)
=
Cov( p,
k)
= = = =
1 1 ✓ ✓ ✓ ✓
1 1 1 1
2
↵ µ ◆2 ⇥ 1 (1 ↵ ◆2 ⇥ 1 (1 ↵ ◆2 ⇥ 1 (1 ↵ ◆2 1 V+ ↵
) (1
2 µ
2)
1
+
(
1
+
1
2)
2 µ
+2 (
1
1
2)
2 µ
+2
2 µ
+ 2)
2) 2 µ
2 µ
+2
2 2 2 v z
+2
V +2 V
2 µ
⇤
Directly,
Cov( p,
⇢pk
p
k)
✓
1
◆
Cov( p, ◆ 1 ⇢pk p k ⇢pa 1 ↵ ✓ ◆ ✓ ◆ 1 1 ⇢pa u 1 ↵ 1 ↵ ✓ ◆ 1 (⇢pk ⇢pa ) 1 ↵ 1
↵ ✓
a)
=
p a
=
p u
=
p µ
=
(1
↵)
✓
1 1
↵
◆2
V
Re-arranging yields the expression in the text. V 2 µ
=
(⇢pk
⇢pa )
p
µ
Correlations:
⇢pk
⇢pa
=
=
r 1 r 1
+ 2 +2
2 +2
V
2 µ
2
2
⇣ ⇣
1+
1+
2 v
2 v
2 µ
2 µ
2 z
2 z
⌘ ⌘ = r⇣
Then, ⇢pk 1 V =1+ 2 ⇢pa µ
1 ⌘2 1 +1+2
1
V
)
14
=
2 µ
⇢pk ⇢pa
1
=
⇢pa ⌘
2 v
2 2 2 v z
2 µ
2 z
⇤
⇤
Given
⇢2pa
)
2 2 v z 2 µ
=
=
Along with the definition of
I.F
2 v
, we then use the expression for ⇢pa to solve for
⇣
⌘2
⌘ ⇢pa
1
1 2⇢2pa
1 2
2 µ
2 z
:
1 2 v
+1+2
✓
2 µ
2 z
◆2 1
⌘ ⇢pa
1 ⌘2 1 ⌘ = + 2 2⇢2pa ⇢pa 2 z
, this allows us to disentangle
1
2 v.
and
Other distortions
Note that distortions have only a second order e↵ect on profits. Therefore, our expressions for stock prices, which use first-order approximations, are una↵ected both under iid and permanent shocks. In other words, the only moments a↵ected by distortions are Cov(p, k) and
k.
With only correlated distortions, both
these moments are scaled by 1 + . Uncorrelated distortions have no e↵ect on Cov(p, k) but increase Transitory shocks.
With correlated distortions, since both Cov(p, k) and
are scaled up by the same
k
factor, ⇢pk remains unchanged:
⇢pk
=
Cov (p, k) p k
=
q
=q 2⇠
4
2⇠2
2 µ
2⇠2
4
2 µ
+
2 2 v z
2 µ
2 µ 2 2 v z
+
2⇠
(
V)
2 u
(1 + ) (
V) (1 + )
2 u
= r⇣
2
1 ⌘⇣ 2 2 1 + v2 z 1
V
2 µ
µ
the same as before. With uncorrelated distortions,
⇢pk
=
Cov (p, k) p k
=
r⇣
=q
2⇠ 4
1 ⌘ ⇣ 2 2 1 + v2 z 1
2⇠2
V
2 µ
µ
so that using (26) will yield an underestimate of
V
2 µ
2 µ
+
2 u
(
⌘=q 2 1 + 2" µ
by an amount
15
2 µ
2 2 v z
2 " 2 µ
.
+
2 "
V)
+
2 " 2 µ
⇢pa V
2 µ
k.
⌘
Permanent shocks.
With correlated distortions, equation (35) becomes Cov ( k, 1+
✓
1 1
↵
◆
p)
⇢pk p 1+ ⇢pk
✓
=
k
1 1
↵
1
=
1
↵
◆
⇢pa
Cov ( p, p µ
+
✓
a) + 1 1
↵
✓
◆2
1 1
↵
◆2
V
V
=
p µ
yielding the same relationship between V and the other moments as before. In other words, using equation (29) leads us to the correct V. If distortions are uncorrelated, we still have ⇢pk
but now
k
=
1 1 ↵
q
2 µ
+2
2 ".
p k
I.G
1 1
↵
◆
⇢pa
p µ
+
✓
1 1
↵
◆2
V
Substituting,
⇢pk
so that now, the formula (⇢kp
=
✓
p
p µ
⇢pa
s
p)
1+2
µ
✓
2 " 2 µ
◆
= ⇢pa
p µ
+
1 1
↵
V
underestimates V.
Correlated signals
Here, we present alternative, more direct derivations of equations (26) and (29), which rely only on conditional normality and not on the correlation structure. Transitory shocks.
We start with the i.i.d. case. Then, pit is an i.i.d. random variable, uncorrelated
with past and future realizations of ait .
Cov ( pit ,
ait )
=
Cov (pit , ait ) + Cov (pit
1 , ait 1 )
= 2Cov (pit , ait )
=
2Cov (pit , Eit ait + $it ) = 2Cov (pit , Eit ait ) + 2Cov (pit , $it )
=
2Cov (pit , Eit ait )
where $it denotes the firm’s forecast error, which is uncorrelated with firm information - in particular, with any element in the firm information set. Therefore, Cov(pit , $it ) = 0. Note that this is true independent of the correlation structure between pit and the other elements of that information set. Now, we use
16
Cov(pit , kit ) =
1 1 ↵ Cov(pit , Eit ait )
Cov ( pit ,
ait )
and divide both sides of the equation by 2Cov (pit , kit ) (1
=
p a
)
↵)
p a
⇢pa ⇢pk
(1
=
↵)
k
=
p a
Cov ( pit ,
k
to get
kit ) (1
p k
↵)
k
a
k
a
Since 2 k
= =
k
)
=
a
✓ ✓
1 1
↵ 1
1 1
1
↵
◆2 ◆2
↵ s
2
2
2
2 µ
(Eit ait ) = ✓
1
✓
◆2
1
2 1 ↵ ◆ ✓ ◆2 V 1 = 2 1 ↵ µ
2 µ 2
V (ait
ait
1)
✓
1
V 2 µ
◆
V
1
2 µ
Combining yields (26). Permanent shocks.
(1
Next, consider the case with ⇢ = 1. Investment now becomes
↵)
kit
pit
+ Eit uit
ait
=
Eit uit + uit 1 Eit 1 uit 1 ⌘ ⇣ 1 ⇣ ˜ it uit = 1 ait 1 + E ait 1 ↵ 1 ↵
=
1
ait
+ Eit
=
2
1 uit 1
1
+ ˜Epit (uit )
⌘
where Epit (uit ) is the expectation conditional on pit alone. Here, we make use of the fact that the marginal investor’s expectation is proportional to Epit . The constant of proportionality ˜ depends on the variance parameters, but, as we will see, will play no role in the determination of V. Stock returns are given by: (1
↵)
pit
=
ait
=
uit
=
1
+ ˜Epit uit
ait
2
˜Ep it
1 uit 1
+ ˜Epit uit ˜Epit 1 uit 1 ⇣ ⌘ ˜Ep uit + 1 ˜ uit 1 + ˜ uit it 1
1
Epit
1 uit 1
Define !it Since !it uit
1
Eit
1
1
⌘ Eit
1 uit 1
Epit
1 uit 1
is in the firm’s information set, it must be orthogonal to the firm’s forecast error $it 1 uit 1 .
1
⌘
We can use this to write the forecast error component in stock returns as the sum of two
17
orthogonal components
uit
Epit
1
1 uit 1
=
(uit
1
=
$it
1
Eit uit + !it
1)
+ !it
1
1
Substituting into the expression for the growth rates, we get
(1
↵) (kit
kit
1)
=
(1
↵) (pit
pit
1)
=
Epit uit + !it + $it 1 ⇣ ⌘ ˜Ep uit + 1 ˜ uit it
1
+ ˜ ($it
1
+ !it
1)
Covariances
Cov ( pit , Cov ( pit ,
ait )
=
kit )
=
= (35)
=
✓
✓
1 1
↵ 1
◆
˜V ar (Ep uit ) it
◆2
˜V ar (Ep uit ) + it
✓
◆2 ⇣
1
1 ↵ 1 ↵ ✓ ◆2 1 ˜V ar ($it 1 ) + 1 ↵ ✓ ◆ ✓ ◆2 ⇣ 1 1 Cov ( p, a) + 1 1 ↵ 1 ↵ ✓ ◆ ✓ ◆2 1 1 Cov ( p, a) + V 1 ↵ 1 ↵
⌘ ˜ Cov (uit
1
✓ ⌘ ˜ V+
1 , $it 1 )
1 1
↵
◆2
˜V
Which gives us
⇢kp 1 1
↵
p k
=
✓
1 1
⇢kp
p µ
=
✓
⇢kp
p µ
=
⇢pa
↵ 1
1
↵
◆ ◆
p µ
⇢pa ⇢pa
+
p
p µ
1 1
µ+
↵
+
✓ ✓
1 1
↵ 1
1
↵
◆2 ◆2
V V
V
Re-arranging yields equation (29).
Appendix II: Data We use annual data on firm-level production variables and stock market returns from Compustat North America (for the US) and Compustat Global (for China and India). For each country, we exclude duplicate observations (firms with multiple observations within a single year), firms not incorporated within that
18
country, and firms not reporting in local currency. We build three year production periods as the average of firm sales and capital stock over non-overlapping 3-year horizons (i.e., K2012 =
K2010 +K2011 +K2012 , 3
and
analogously for sales). We measure the capital stock using gross property, plant and equipment (PPEGT in Compustat terminology), defined as the “valuation of tangible fixed assets used in the production of revenue.” We then calculate investment as the change in the firm’s capital stock relative to the preceding period. We construct the firm fundamental ait as the log of value-added less the relevant ↵ (which depends on the case) multiplied by the log of the capital stock. We compute value-added from revenues using a share of intermediates of 0.5 (our data does not generally include measures of intermediate inputs). Finally, we first-di↵erence the investment and fundamental series to compute investment growth and changes in fundamentals. Stock returns are constructed as the change in the firm’s stock price over the three year period, adjusted for splits and dividend distributions. We follow the procedure outlined in the Compustat manual. In Compustat terminology, and as in the text, using pit as shorthand for log returns, returns for the US are computed as pit = log
✓
PRCCMit ⇤ TRFMit AJEXMit
◆
log
✓
PRCCMit 1 ⇤ TRFMit AJEXMit 1
1
◆
where periods denote 3 year spans (i.e., returns for 2012 are calculated as the adjusted change in price between 2009 and 2012), PRCCM is the firm’s stock price, and TRFM and AJEXM adjustment factors needed to translate prices to returns from the Compustat monthly securities file. Data are for the last trading day of the firm’s fiscal year so that the timing lines up with the production variables just described. The calculation is analogous for China and India, with the small caveat that global securities data come daily, so that the Compustat variables are PRCCD, TRFD, and AJEXDI, where “D” denotes days. Again, the data are for the last trading day of the firm’s fiscal year. Table A.1 displays summary statistics for 2012: the number of firms in each country, the share of GDP they account for, and average sales and capital stock in US dollars. GDP data is from the World Bank Development Indicators database. Foreign currencies are converted to US dollars using the exchange rate on December 31, 2012. To extract the firm-specific variation in our variables, we regress each on a time fixed-e↵ect and work with the residual. This eliminates the component of each series common to all firms in a time period and leaves only the idiosyncratic variation. As described in the text, we limit our sample to a single cross-section, namely 2012, and finally, we trim the 2% tails of each series. It is then straightforward to compute the target moments, i.e.,
2 p,
⇢pi , and ⇢pa . As described in the text, we lag returns by one period, so that, e.g., ⇢pi is
the correlation of 2006-09 returns with investment growth from 2009-12. To estimate the parameters governing the evolution of firm fundamentals, i.e., the persistence ⇢ and
19
Table A.1 Sample Statistics No. of firms
Share of GDP
Avg. Sales ($M)
Avg. PPEGT ($M)
6618 2270 2028
0.42 0.12 0.21
2792 851 388
2117 639 323
US China India
variance of the innovations
2 µ,
we perform the autoregression implied by (1). Here we use annual observations
on ait at a 3-year frequency in order to simplify issues of time aggregation. We estimate the process using our data from 2012 and 2009. We include a time fixed-e↵ect in order to isolate the idiosyncratic component of the innovations in ait . Di↵erences in firm fiscal years means that di↵erent firms within the same calendar year are reporting data over di↵erent time periods, and so the time fixed-e↵ect incorporates both the reporting year and month. The results from this regression deliver an estimate for ⇢ and
2 µ,
from which we trim the
1% tails. Our leverage adjustment is as follows: we assume that claims to firm profits are sold to investors in the form of both debt and equity in a constant proportion (within each country). This implies that the payo↵ from an equity claim is Sit = Vit
Dit
1,
where Vit is the value of the unlevered firm and Dit
1
= dEt [Vit ],
where d 2 (0, 1) represents the share of expected firm value in the hands of debt-holders. In other words, firm value is allocated to investors as a debt claim that pays o↵ a constant fraction of its ex-ante expected value and as a residual claim to equity holders. The change in value of an equity claim is then equal to dividing both sides by the ex-ante expected value of the claim (i.e., the price) S = V gives returns as
Sit S
=
Vit . (1 d)V
Taking logs and computing variances shows
volatility in (unlevered) firm value is a fraction (1
2 Vit
Sit =
Vit and
dV, where V¯ = Et [Vit ], = (1
d)
2
2 sit ,
i.e., the
2
d) of the volatility in (levered) equity returns. To
assign values to d in each country, we examine the debt-asset and debt-equity ratios of the set of firms in Compustat over the period 2006-2009. Because these vary to some degree from year to year and depend to some extent on the precise approach taken (i.e., whether we use debt-assets or debt-equity and whether we compute average ratios or totals), we simply take the approximate midpoints of the ranges for each country, which are about 0.30 for the US and India and 0.16 for China, leading to adjustment factors of about 0.5 and 0.7, respectively.
20
Appendix III: Quantitative Exercises III.A
Case 1 vs Case 2
Table A.2 reports the moments and parameters using labor data for US firms (measured as total employment), rather than capital. Moments are constructed as in the baseline analysis, using employment rather than capital. We denote the correlation of stock returns with changes in employment growth ⇢pn . Table A.2 Moments and Parameters - Labor Target moments
III.B
⇢pn
⇢pa
0.17
0.17
Parameters
2 p
⇢
0.24
µ
0.87
e
0.40
v
0.29
z
0.41
2.91
Adjustment Costs
When firms are fully informed but subject to capital adjustment costs, the dynamic optimization problem of the firm is characterized by the value function: ⇣ V A˜it , Kit
1
⌘
↵ ˜ = max GA˜it Kit Kit ,Nit
Kit
⇣Kit
1
✓
◆2 ⇣ 1 + EV A˜it+1 , (1
Kit Kit 1
) Kit
⌘
To characterize G and A˜it , we start with the firm’s labor choice problem
max Pit Yit Nit
1
↵1 ↵2 W Nit = max Yt ✓ Ait Kit Nit Nit
W Nit
where ↵’s are as defined in (2). Optimizing over Nit and substituting back into the objective gives
Pit Yit
W Nit = (1
↵2 )
⇣ ↵ ⌘ 1 ↵↵2 2
2
W
1
Yt ✓ 1
1 ↵2
↵ ˜ A˜it Kit
where ↵ ˜ and A˜it are defined as in case 2 in the text. We can then solve for
G = (1
↵2 )
h⇣ ↵ ⌘↵2 2
W
21
1
Yt ✓
i1
1 ↵2
Next, using the firm’s labor optimality condition, labor market clearing implies ⇣↵ ⌘1 2
1 ↵2
W
1
Yt ✓ 1
1 ↵2
Z
↵ ˜ A˜it Kit di = N
from which we can solve for ⇣ ↵ ⌘↵ 2 2
W
1 ✓
Yt =
Z
↵ ˜ A˜it Kit
and so G = (1
↵2 )
Z
1 ↵2 ✓ 1
(1 ✓↵2 )
di
N✓
1 ✓
↵ ˜ A˜it Kit di
✓↵2 1
N✓
✓
✓
1 ↵2 (1
↵2 )
1 ↵2
Note that under full information, the production side of the economy is completely decoupled from stock markets, so we can now solve the model laid out above. Starting with a guess for the general equilibrium term G, we solve for the value functions (using a standard iterative procedure and a discretized grid for capital), simulate to obtain the steady state distributions and verify that our initial guess for G is consistent with that distribution. If not, we update the guess and iterate until convergence. We then solve for stock prices, again using the conjecture that the informed investors follow a threshold rule. Proceeding exactly as in the baseline model, we can then derive the following functional equation for the price:
P (ait
1 , kit 1 , ait , zit )
=
Z
⇡ (ait , kit 1 ) H (ait |ait 1 , ait + v zit , Pit ) Z + P¯ (ait , k ⇤ (ait , kit 1 )) H (ait |ait 1 , ait +
v zit , Pit )
The model is parameterized as described in the text. Table A.3 reports the full set of target moments and parameter estimates. Notice that the first two columns are identical to those in Table II (case 2). denotes the variance of investment rates, which is used to pin down the size of the adjustment cost ⇣. Table A.3 Targets and Parameters - Adjustment Cost Model Target moments ⇢pa US China India
0.18 0.06 0.08
2 p
0.23 0.14 0.23
22
2 k
0.10 0.15 0.10
Parameters ⇣ 0.45 0.18 0.52
v
0.37 1.00 0.69
z
4.25 4.00 6.36
2 k
III.C
Measurement Error
With measurement error in revenues, the following equations relate the observed variables and moments (denoted with a hat) with their true counterparts (no hat). In this case, the a↵ected moments are ⇢,
2 µ,
and ⇢pa .
⇢b =
µ bit
) bµ2
⇢bpa
Cov ait + ✏yit , ait 1 + ✏yit 2 2 a + ✏y
= b ait
⇢bb ait
1
=b ait
=
ait + ✏yit
=
µit + ✏yit + (⇢
⇢ait
=
2 µ
+
2 ✏y
+ (⇢
=
2 µ
+
2 ✏y
+
=
2 µ
+
2 ✏y
= =
+
⇢b ait ⇢✏yit
1
⇢b) ait 2
⇢b) !2
2 ✏y 2 a
2 ⇢b2 ✏y
⇢b2
1+
Cov ( pit , b ait b ait b p a s 2 2 b a 2 ✏y ⇢pa b2 a
1 1
1)
1
=⇢
1
+ (⇢
⇢b) ✏yit
2 ✏y
ba2
!
1
2 ✏y
2 ✏y
=
ba2
2 ✏y
+ ⇢b2
2 b a
1)
1
⇢b) ait
+ (⇢
+ ⇢b2 2 a
⇢b) b ait
+ (⇢
⇢b✏yit
1 2 a
Cov (ait , ait = 2 2 a + ✏y
1
2 ✏y
!
Cov ( pit , ait p
a
ait
1)
a
b
a
The corrections to the observed moments when capital is measured with error are similar and omitted for brevity. Table A.4 reports the moments and parameters after adjusting for measurement error for the cases in Table X.
III.D
Sectoral Analysis
Our sectoral analysis in Section V.A makes use of data from I/B/E/S Guidance and Compustat. We obtain from I/B/E/S data on firm forecasts of annual revenues and the corresponding realizations. We examine forecasts that are made at approximately a 1 year horizon, specifically, between 10 and 14 months prior to the reporting date. There are very few forecasts available at longer horizons. For forecasts that include a range, we take the midpoint. In order to associate I/B/E/S firms with industry codes and additionally to use a consistent sample of firms from the two sources, we merge the I/B/E/S data with Compustat. We are then able to place each I/B/E/S firm into an SIC code. We examine only firms who are present both in I/B/E/S and Compustat. In this way, we are able to use exactly the same set of firms to construct the sectoral indicator of uncertainty from I/B/E/S and to estimate V from Compustat. We include the period
23
Table A.4 Adjusted Moments and Parameters - Measurement Error Target moments 2 p
Parameters
⇢pi
⇢pa
⇢
Case 2 - 10% error in y US China India
0.23 0.16 0.25
0.22 0.07 0.09
0.23 0.14 0.23
0.92 0.79 0.94
0.42 0.48 0.51
0.27 0.58 0.74
0.22 0.70 0.85
4.89 3.83 2.65
Case 2 - 25% error in y US China India
0.23 0.16 0.25
0.36 0.10 0.14
0.23 0.14 0.23
0.94 0.82 0.95
0.38 0.45 0.48
0.10 0.47 0.63
0.17 0.45 0.22
3.56 4.99 8.00
Case 2 - 10% error in k US China India
0.35 0.22 0.32
0.20 0.06 0.08
0.23 0.14 0.23
0.92 0.78 0.94
0.44 0.50 0.52
0.59 0.77 1.12
0.33 0.78 0.84
3.52 3.45 3.06
Case 1 - 10% error in y US China India
0.24 0.16 0.26
0.11 0.02 0.00
0.23 0.14 0.23
0.89 0.77 0.89
0.44 0.51 0.53
0.58 0.78 1.72
0.57 1.11 1.60
3.23 3.81 4.80
Case 1 - 25% error in y US China India
0.24 0.16 0.26
0.18 0.03 0.01
0.23 0.14 0.23
0.91 0.80 0.90
0.40 0.48 0.49
0.35 0.81 1.47
0.34 0.91 1.47
3.47 4.80 4.53
Case 1 - 10% error in k US China India
0.37 0.22 0.33
0.11 0.02 0.00
0.23 0.14 0.23
0.88 0.77 0.89
0.44 0.52 0.53
1.26 1.03 2.53
0.63 1.14 1.70
3.11 3.80 4.41
µ
e
v
z
1998-2013. I/B/E/S data prior to 1998 is sparse. Once the sample is constructed, we group firms into 2-digit SIC industry groups and work on a sectorby-sector basis. Sectoral classifications are listed in Table A.5, along with the associated SIC codes and the number of firm-year observations from each of our data sources. The set of firms is the same across the samples; the number of firm-year observations di↵ers since there may not be forecasts for each firm-year observation in Compustat, and vice versa, there may not be sufficient data from Compustat to include a firm-year observation despite having a forecast from I/B/E/S. There are two sector groups where we do not have sufficient data from I/B/E/S - Agriculture, Forestry, Fishing, and Mining. Within each sector, for each forecast-realization pair from I/B/E/S, we compute the log forecast error as ln (forecast)
ln (actual). To
extract the idiosyncratic component of the forecast error, we regress the errors on a time fixed-e↵ect and work with the residual. We trim the 1% tails of the errors and compute the variance of the forecast errors, i.e., the mean squared forecast error. This is our indicator of uncertainty at the sectoral level. Compustat 24
moments are computed following the same procedure as for our baseline analysis. The moments are now constructed on a sector-by-sector basis and over the period 1998-2013 where we eliminate time-specific factors (e.g., inflation) by extracting time fixed-e↵ects from all series. Table A.5 reports the mean squared error of firm forecasts from I/B/E/S, the empirical moments from Compustat, the resulting parameter estimates, and our estimate of V, all on a sectoral basis.
III.E
Time-Series Analysis
We construct the empirical moments in our time-series analysis following the same procedure as for our baseline. Moments are computed year-by-year from 1998 to 2012 on a balanced panel of firms which have the necessary data for all years. Because calculation of the moments in each year require 9 years worth of data, firms must have data back to 1990 to be included. There are about 785 firms in our final sample. Table A.6 reports the empirical moments and estimated parameter values for each year.
25
26
Retail Trade FIRE Services Construction Manufacturing Trans and PU Wholesale Trade
52-59 60-69 70-89 15-19 20-39 40-49 50-51
SIC Range 962 211 2378 50 6220 366 269
Compustat 516 397 2357 102 4502 361 225
I/B/E/S
No. of Observations 0.004 0.007 0.011 0.014 0.014 0.015 0.015
MSE 0.35 0.17 0.28 0.36 0.30 0.37 0.10
⇢pi 0.18 0.18 0.23 0.35 0.17 0.12 0.33
⇢pa 0.36 0.34 0.53 0.68 0.38 0.44 0.24
2 p
Target Moments 0.90 0.88 0.81 0.69 0.79 0.77 0.96
⇢
Sectoral Uncertainty - Moments and Parameters
Table A.5
0.19 0.36 0.35 0.48 0.36 0.33 0.23
µ
0.29 0.26 0.29 0.30 0.42 0.76 0.13
e
v
0.09 0.43 0.20 0.11 0.29 0.30 0.61
Parameters 5.92 2.17 4.13 6.98 3.38 3.79 0.50
z
0.023 0.042 0.045 0.057 0.069 0.085 0.011
V
Table A.6 Time-Series of Uncertainty - Moments and Parameters Target moments 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
⇢pi
⇢pa
0.28 0.19 0.27 0.32 0.28 0.28 0.21 0.27 0.23 0.32 0.30 0.27 0.26 0.20 0.21
0.20 0.15 0.12 0.09 0.13 0.12 0.13 0.17 0.10 0.10 0.14 0.14 0.18 0.12 0.15
2 p
0.29 0.27 0.27 0.33 0.38 0.47 0.36 0.42 0.41 0.26 0.27 0.22 0.25 0.36 0.32
Parameters ⇢
µ
0.85 0.91 0.93 0.91 0.88 0.95 0.95 0.95 0.93 0.95 0.97 0.97 0.95 0.90 0.97
0.34 0.35 0.33 0.33 0.34 0.34 0.32 0.29 0.28 0.29 0.31 0.32 0.29 0.27 0.27
e
0.32 0.33 0.40 0.50 0.47 0.49 0.34 0.30 0.35 0.41 0.40 0.40 0.33 0.28 0.28
v
0.35 0.79 1.12 0.62 0.36 0.45 0.71 0.39 0.47 1.38 1.54 1.62 1.45 0.36 0.74
z
2.22 0.94 0.63 1.83 3.40 2.98 1.29 2.02 2.45 0.46 0.40 0.43 0.34 2.77 0.86
V
V Detrended
0.050 0.052 0.057 0.071 0.073 0.073 0.050 0.040 0.046 0.048 0.052 0.056 0.040 0.037 0.035
-0.013 -0.010 -0.003 0.012 0.016 0.018 -0.003 -0.012 -0.004 0.000 0.004 0.011 -0.004 -0.005 -0.006
References Hsieh, C. and P. Klenow (2009): “Misallocation and Manufacturing TFP in China and India,” Quarterly Journal of Economics, 124, 1403–1448.
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