Macroeconometrics of Investment and the User Cost of Capital Presentation Sample

Sample Presentation

Macroeconometrics of
Investment and the User Cost
of Capital
Thethach Chuaprapaisilp

April 14, 2009

Thethach Chuaprapaisilp: Macroeconometrics of Investment and the User Cost of Capital, 1

Sample Slides

Aim: Estimate the long-run user cost elasticity of business
investment by using a general equilibrium macroeconometric
model and cointegration technique.
The Jorgenson (1963) ‘user cost of capital’ as the real rental
price of capital services or the costs of holding capital:
Ct ≡ Pt−1 rt + δt PtI − PtI − Pt−1
I I

(Interest/opportunity cost + Depreciation cost - Capital gain)
Long-run neoclassical investment equation from the FOC of
the optimal capital accumulation problem
∞
max{Kt−1 ,Lt ,It } V = (1 + r )−t PtY Qt − wt Lt − PtI It
t=0
s.t. Ft (Kt−1 , Lt ) = Qt and ∆Kt = It − δKt−1
implies

PtY FK (Kt−1 , Lt ) = (1 + r ) Pt−1 − (1 − δ) PtI (≡ Ct )
I


Sample Slides

I PtI − Pt−1
I
Ct ≡ Pt−1 r + δ − (1 − δ) I
.
Pt−1

Can also obtain from a durable goods model of production as
in Jorgenson and Yun (1991) and Hall and Jorgenson (1967)
with varying weighted average of rates of return on debt and
equity, corporate tax rate, investment tax credit and
depreciation allowances.
Normalizing investment goods price by the price of output,
PtY and assuming beginning-of-period gross investment
(It = ∆Kt+1 + δt Kt ),

PtI Pt+1 − PtI
I
1 − ITCt − τt zt
CtK ≡ rtl + δt − Et .
PtY PtI 1 − τt


Sample Slides

Assuming a CES production function, Yt = (aKtσ + bLσ )1/σ
t
the ﬁrst-order condition gives the long-run investment
equation (in logs),

1 1
kt = ln a + yt − ck
1−σ 1−σ t

kt = α0 + yt + αR Rt , Rt ≡ ctk
Many studies using US, UK and Canadian data obtain αR of
around -0.4 (Chirinko et al., 2007; Ellis and Price, 2004) to
-0.7 for total private non-residential capital stock and close to
one for equipment capital (-0.9 in Caballero, 1994; -1.6 in
Schaller, 2006).
Tevlin and Whelan (2003): -1.59 for computers, -0.13 for
noncomputing equipments and -0.18 for equipment capital in
total.


Figure: 1 U.S. data. All variables are in logs. kt is the log of capital stock for non-residential business sector
equipment and software calculated from seasonally adjusted real investment series using the perpetual inventory
method and depreciation smoothing technique of Diewert (2008). Rt is the log of the user cost of capital with the
corresponding implicit price index as the purchase price of investment goods. Rt includes after-tax real financial
cost of capital for producers’ durable equipment (a weighted average of 5-year Treasury bond yield and Moody’s
AAA corporate bond rate plus risk premia less expected inflation over 30 years) and tax variables from the FRB/US
model. The log short-term real return measure, rrpt is calculated based on the investment goods price inflation and
I I Y
multiplied by the relative investment goods price so that rrpt = [ln(1 + rt ) − πt+1 ] × Pt /Pt where rt is the
3-month CD rate net of federal average marginal income tax on interest received.

Figure: 2 U.S. data in ﬁrst diﬀerences.

Sample Slides

Estimate αR using a vector error correction model (VECM) of
a macroeconomy here assuming a tendency for output and
real interest rates to move towards their long-run natural
(ﬂexible price equilibrium) values as represented by
cointegrating relationships.
Follow the structural cointegrated VAR methodology of
Garratt, Lee, Pesaran and Shin (2006) and Juselius (2006).
Expand the two-equation VECM of Ellis and Price (2004) to a
general equilibrium system.
Long-run counterpart of the macromonetary framework by
Gali and Gertler (2007). Concentrate on the long run and use
the cost of capital instead of Tobin’s q here.


Long-run reduced form equilibrium conditions provide six cointegrating relations,
lt = b10 − b11 t + yt − β25 ωt + ξ1,t+1 Ld
kt = b20 + yt − β25 Rt + ξ2,t+1 Investment
yt = b30 + b31 t + β31 lt + β34 kt + ξ3,t+1 LRAS
yt = b40 + β44 kt + β46 ct + β47 nxt + β49 gt + ξ4,t+1 IS
ct = b50 + β51 lt + β51 ωt + β54 kt − β58 rrt + ξ5,t+1 Consumption
Rt = b60 + b68 rrt + ξ6,t+1 YC
zt = (lt , yt , wt , kt , Rt , ct , nxt , rrt , gt ) = (yt , gt )
ξt = β zt−1 − b0 − b1 (t − 1)
p−1
∆zt = a0 − αξt + Γi ∆zt−i + vt
i=1
p−1
∆zt = a0 − α β zt−1 − b0 − b1 (t − 1) + Γi ∆zt−i + vt
i=1
p−1
∆zt = a + bt − Πzt−1 + Γi ∆zt−i + vt
i=1
where a = a0 + α (b0 − b1 ), b = αb1 and Π = αβ .
∆yt = ay − Π∗ z∗ + Γy 1 ∆zt−1 + ψ y 0 ∆gt + uyt ,
t−1

z∗ = (zt−1 , t) = (yt−1 , gt−1 , t) and Π∗ = αβ ∗ , β ∗ = (β , −b1 ).
t−1
r0t = Π∗ r1t + εt

Table: 6.1 Long-run β∗ estimates for the over-identified VECM with rank(Π∗ ) = 5
and p = 2, 1962q4 to 2006q2.

z∗
t−1 Labor Investment Production IS Consumption

lt 1.0000 −0.85208∗ −0.45304∗
(0.037352)
yt −1.0000 −1.0000 1.0000 1.0000

wt 0.12922∗ −0.45304∗
(0.032059) (0.056240)
kt 1.0000 −0.26361∗ −0.035386 −0.40647∗
(0.018388) (0.013755) (0.024968)
∗
Rt 1.9125 −0.22315∗
(0.11895) (0.017498)
ct −0.77822∗ 1.0000
(0.026293)
nxt −0.067944∗ −1.0111∗ −0.049279∗
(0.0093073) (0.18595) (0.0044202)
rrtp −0.26586∗ −6.6812∗ 0.23153∗ −0.10696∗ 0.77444∗
(0.057164) (1.0655) (0.050414) (0.019996) (0.091525)
gt 0.073148∗ −0.18590∗ −0.16177∗ −0.23659∗
(0.025286) (0.025986) (0.012769) (0.033603)
Trend 0.0028339∗ 0.00065454
(0.00013614) (0.00030597)

United States data. Standard errors in parentheses. Estimated parameters in bold indicate significance of the t-ratios
at the 5% level and at 1% level with an asterisk. Likelihood function maximized under general restrictions based on
Doornik (1995) (scaled linear) switching algorithm under weak convergence criterion of |l (θi+1 ) − l (θi )| ≤ = 0.005.
Log-likelihood = 5247.98759. −T /2 ln |Ω| = 7234.50154. Beta is identified and 2 over-identifying restrictions are imposed.
LR test statistics: χ2 (2) = 3.0193 [0.2210]. Number of observations: 175. Number of parameters: 151. The log short-term
real return measure is calculated based on the investment goods price inflation and multiplied by the relative investment
goods price so that rrtp = [ln(1 + rt ) − πt+1 ] × PtI /PtY .
I

Table: 6.2 α adjustment coefficient estimates for the over-identified VECM with
rank(Π∗ ) = 5 and p = 2, 1962q4 to 2006q2.

Equation Labor Investment Production IS Consumption

∆lt −0.066433 0.0095344∗ 0.028934 −0.11145 0.090029
(0.10738) (0.0043023) (0.13117) (0.093880) (0.052082)

∆yt 0.49876∗ −0.010069 0.47283∗ −0.48830∗ −0.088506
(0.15039) (0.0060257) (0.18371) (0.13149) (0.072945)

∆wt 0.078638 −0.0027374 −0.0051453 0.010466 0.017234
(0.071941) (0.0028824) (0.087881) (0.062897) (0.034894)

∆kt 0.077271∗ −0.0028891∗ 0.087255∗ 0.0050440 −0.012648
(0.022138) (0.00088701) (0.027043) (0.019355) (0.010738)

∆Rt −1.9667∗ 0.012601 −3.9010∗ 1.9932∗ 1.3729∗
(0.71329) (0.028579) (0.87134) (0.62362) (0.34597)

∆ct 0.34074∗ −0.0035532 0.35063 −0.0013989 −0.091877
(0.14455) (0.0057918) (0.17658) (0.12638) (0.070114)

∆nxt 0.73140 0.039095 −0.19918 0.91785 0.83582∗
(0.70116) (0.028093) (0.85651) (0.61301) (0.34008)

∆rrtp −0.077447 0.0058118 −0.12361 0.40675 −0.088550
(0.27242) (0.010915) (0.33278) (0.23818) (0.13213)

United States data. Standard errors in parentheses. Estimated parameters in bold indicate significance of the t-ratios
at the 10% level and at 5% level with an asterisk.

Sample Slides

Long run β ∗ and α estimates obtained from the concentrated
model: r0t = Π∗ r1t + εt of the conditional VECM:
∆yt = ay − Π∗ z∗ + Γy 1 ∆zt−1 + ψ y 0 ∆gt + uyt where
t−1
Π∗ = αβ ∗ .
ˆ
Estimated cointegration relations, β ∗ r1 from Table 6.1
correspond to the following reduced form error correction
ˆ∗ ˆ
terms ξ t = β ∗ z∗ :
t−1

ξ1,t+1 = lt − yt + 0.1292ωt − 0.0679nxt − 0.2659rrtp + 0.0731gt + 0.0028t
ˆ Labor
ξ2,t+1 = kt − yt + 1.9125Rt − 1.0111nxt − 6.6812rrtp
ˆ Invest
ˆ
ξ3,t+1 = yt − 0.8521lt − 0.2636kt + 0.2315rrtp − 0.1859gt + 0.0007t Product
ξ4,t+1 = yt − 0.0354kt − 0.7782ct − 0.0493nxt − 0.1070rrtp − 0.1618gt
ˆ IS
ξ5,t+1 = ct − 0.4530lt − 0.4530ωt − 0.4065kt − 0.2231Rt + 0.7744rrtp − 0.2366gt
ˆ Consump


ˆ
Figure: 6. Reduced form errors for the long-run relations. Estimated −ξt for
the over-identiﬁed VECM with rank(Π∗ ) = 5 and p = 2, 1962q4 to 2006q2.

Figure: 8. Generated quarterly stocks of business R&D in billions of chained
(2000) dollars, 1960-2004 compared with the BEA’s annual capital stock data.

Figure: 9. Calculated quarterly stock series for business R&D in billions of
chained (2000) dollars, 1982-2004 from the poisson model with diﬀerent λ
values compared with that from the depreciation smoothing procedure.

ˆ
Figure: 10. Reduced form errors for the long-run relations. Estimated −ξt for
VECM with R&D capital stock, dt included in the investment relation,
rank(Π∗ ) = 5 and p = 2, 1962q4 to 2004q4.

Table: 16. Summary of elasticity estimates.
Table User Cost Wage Variables

Rank 5 VECM with net exports:

(6.1) U.S. data -1.9125 -0.1292 y , k, R, nx, rr
[-1, 1, 1.9125, -1.0111, -6.6812]

(12.1) Canadian data -1.0237 0.1811 y , k, R, rr
[-1, 1, 1.0237, -5.7059]

Rank 4 VECM without net exports:

(8) E&S investment -0.4587 -0.4207 y , k, R, rr , g , Trend
[-1, 1, 0.4587, -1.3768, 0.4990, -0.0070]

(10) Non high-tech investment -1.2633 -0.2153 y , k o , R o , rr o
[-1, 1, 1.2633, -8.6959]

Dynamic OLS:

U.S. E&S investment -0.7 to -1.1
U.S. non high-tech investment -0.3 to -0.5
Canadian E&S investment -0.9 to -1.1

United States estimates for equipment and software (E&S) investment unless stated otherwise. Variables are
those included in the investment relation with the corresponding cointegration vector at the bottom. The wage
coefficient in the Canadian VECM in Table 12.1 has an opposite sign but is not significantly different from zero.

Table: 16. Summary of elasticity estimates (continued).
Table User Cost Wage Variables

Rank 5 VECM with R&D capital stock:

(14.1) Neutral R&D technology -0.4870 -0.2858 y , k, R, rr , g , Trend
[-1, 1, 0.4870, -1.5182, 0.4992, -0.0069]

(15.1) Capital augmenting R&D tech. -0.2768 -0.2020 y , k, R, rr , d
[-1, 1, 0.2768, -2.1942, -0.3904]

Dynamic OLS:

U.S. E&S investment -0.7 to -1.1
U.S. non high-tech investment -0.3 to -0.5
Canadian E&S investment -0.9 to -1.1

Smaller VECM:

(17) Canadian data with investment -1.3578 y , k, R
[-1, 1, 1.3578]

(18) U.S. data single equation -0.2543 y , k, R, Trend
[-1, 1, 0.2543, -0.0052]

United States estimates for equipment and software (E&S) investment unless stated otherwise. Variables are
those included in the investment relation with the corresponding cointegration vector at the bottom.

Conclusion

According to the estimation results for α adjustment
coefficients, the user cost of capital is found to be statistically
significantly adjusting to the long-run reduced form shocks on
several cointegration relations.
User cost variable is endogenous to the general equilibrium
system and responses to macroeconomic shocks including
external and policy shocks.
Net exports is also endogenous to the system and needs to be
included in the investment relation for cointegration.
Influence of domestic savings and investment on the
equilibrium long-term real interest hence on the endogenous
user cost of capital in a large open economy. Large estimated
user cost elasticities estimates may be biased due to the
equilibrium current account adjustments (S − I = NX ).


Conclusion

Small user cost elasticities for the U.S. obtained in the VECM
without net exports but with linear time trend (close to -0.4 in
Chirinko et al., 2007) or with R&D capital-augmenting
technology dt (close to the wage elasticity of -0.25 in
Juselius M., 2008).
Endogeneity of the user cost that is inﬂuenced by the demand
and supply sides of a large economy hence the domestic
supply of and demand for capital.
User cost elasticity estimate of around -1.0 for Canadian
VECM. In a small open economy the interest rates and capital
goods prices are largely predetermined internationally and
much less aﬀected by the domestic demand for and supply of
capital.
Perfectly elastic supply of capital from abroad so easier to
identify domestic capital demand in estimation
(Schaller, 2006; Coulibaly and Millar, 2007).


Macroeconometrics of Investment and the User Cost of Capital Presentation Sample

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