Macroeconometrics of Investment and the User Cost of Capital Presentation Sample

Sample Presentation

Macroeconometrics ofInvestment 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

I Aim: Estimate the long-run user cost elasticity of businessinvestment by using a general equilibrium macroeconometricmodel and cointegration technique.

I The Jorgenson (1963) ‘user cost of capital’ as the real rentalprice of capital services or the costs of holding capital:

Ct ≡ P It−1rt + δtP

It −

(P I

t − P It−1

)(Interest/opportunity cost + Depreciation cost - Capital gain)

I Long-run neoclassical investment equation from the FOC ofthe optimal capital accumulation problem

maxKt−1,Lt ,It V =∞∑

t=0(1 + r)−t

[PY

t Qt − wtLt − P It It]

s.t. Ft(Kt−1, Lt) = Qt and ∆Kt = It − δKt−1

implies

PYt FK (Kt−1, Lt) = (1 + r) P I

t−1 − (1− δ) P It (≡ Ct)


Sample Slides

Ct ≡ P It−1

r + δ − (1− δ)

[P I

t − P It−1

P It−1

].

I Can also obtain from a durable goods model of production asin Jorgenson and Yun (1991) and Hall and Jorgenson (1967)with varying weighted average of rates of return on debt andequity, corporate tax rate, investment tax credit anddepreciation allowances.

I Normalizing investment goods price by the price of output,PY

t and assuming beginning-of-period gross investment(It = ∆Kt+1 + δtKt),

CKt ≡

P It

PYt

r lt + δt − Et

(P I

t+1 − P It

P It

)(1− ITCt − τtzt

1− τt

).


Sample Slides

I Assuming a CES production function, Yt = (aKσt + bLσt )1/σ

the first-order condition gives the long-run investmentequation (in logs),

kt =1

1− σln a + yt −

1

1− σckt

kt = α0 + yt + αRRt , Rt ≡ ckt

I Many studies using US, UK and Canadian data obtain αR ofaround -0.4 (Chirinko et al., 2007; Ellis and Price, 2004) to-0.7 for total private non-residential capital stock and close toone for equipment capital (-0.9 in Caballero, 1994; -1.6 inSchaller, 2006).

I Tevlin and Whelan (2003): -1.59 for computers, -0.13 fornoncomputing equipments and -0.18 for equipment capital intotal.


Figure: 1 U.S. data. All variables are in logs. kt is the log of capital stock for non-residential business sectorequipment and software calculated from seasonally adjusted real investment series using the perpetual inventorymethod and depreciation smoothing technique of Diewert (2008). Rt is the log of the user cost of capital with thecorresponding implicit price index as the purchase price of investment goods. Rt includes after-tax real financialcost of capital for producers’ durable equipment (a weighted average of 5-year Treasury bond yield and Moody’sAAA corporate bond rate plus risk premia less expected inflation over 30 years) and tax variables from the FRB/USmodel. The log short-term real return measure, rrpt is calculated based on the investment goods price inflation and

multiplied by the relative investment goods price so that rrpt = [ln(1 + rt )− πIt+1]× P I

t /PYt 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

I Estimate αR using a vector error correction model (VECM) ofa macroeconomy here assuming a tendency for output andreal interest rates to move towards their long-run natural(flexible price equilibrium) values as represented bycointegrating relationships.

I Follow the structural cointegrated VAR methodology ofGarratt, Lee, Pesaran and Shin (2006) and Juselius (2006).Expand the two-equation VECM of Ellis and Price (2004) to ageneral equilibrium system.

I Long-run counterpart of the macromonetary framework byGali and Gertler (2007). Concentrate on the long run and usethe cost of capital instead of Tobin’s q here.


Long-run reduced form equilibrium conditions provide six cointegrating relations,

lt = b10 − b11t + yt − β25ωt + ξ1,t+1 Ld

kt = b20 + yt − β25Rt + ξ2,t+1 Investment

yt = b30 + b31t + β31lt + β34kt + ξ3,t+1 LRAS

yt = b40 + β44kt + β46ct + β47nxt + β49gt + ξ4,t+1 IS

ct = b50 + β51lt + β51ωt + β54kt − β58rrt + ξ5,t+1 Consumption

Rt = b60 + b68rrt + ξ6,t+1 YC

zt = (lt , yt ,wt , kt ,Rt , ct , nxt , rrt , gt)′ = (y′t , gt)′

ξt = β′zt−1 − b0 − b1(t − 1)

∆zt = a0 −αξt +

p−1∑i=1

Γi ∆zt−i + vt

∆zt = a0 −α[β′zt−1 − b0 − b1(t − 1)

]+

p−1∑i=1

Γi ∆zt−i + vt

∆zt = a + bt −Πzt−1 +

p−1∑i=1

Γi ∆zt−i + vt

where a = a0 +α (b0 − b1), b = αb1 and Π = αβ′.

∆yt = ay −Π∗z∗t−1 + Γy1∆zt−1 +ψy0∆gt + uyt ,

z∗t−1 = (z′t−1, t)′ = (y′t−1, gt−1, t)′ and Π∗ = αβ′∗, β′∗ = (β′,−b1).

r0t = Π∗r1t + εt

Table: 6.1 Long-run β∗ estimates for the over-identified VECM with rank(Π∗) = 5and 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)rrp

t −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-ratiosat the 5% level and at 1% level with an asterisk. Likelihood function maximized under general restrictions based onDoornik (1995) (scaled linear) switching algorithm under weak convergence criterion of |l (θi+1)− l (θi )| ≤ ε = 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-termreal return measure is calculated based on the investment goods price inflation and multiplied by the relative investmentgoods price so that rrp

t = [ln(1 + rt)− πIt+1]× P I

t /PYt .

Table: 6.2 α adjustment coefficient estimates for the over-identified VECM withrank(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)

∆rrpt −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-ratiosat the 10% level and at 5% level with an asterisk.

Sample Slides

I Long run β∗ and α estimates obtained from the concentratedmodel: r0t = Π∗r1t + εt of the conditional VECM:∆yt = ay −Π∗z∗t−1 + Γy1∆zt−1 +ψy0∆gt + uyt whereΠ∗ = αβ′∗.

I Estimated cointegration relations, β′∗r1 from Table 6.1

correspond to the following reduced form error correctionterms ξ

∗t = β

′∗z∗t−1:

ξ1,t+1 = lt − yt + 0.1292ωt − 0.0679nxt − 0.2659rrpt + 0.0731gt + 0.0028t Labor

ξ2,t+1 = kt − yt + 1.9125Rt − 1.0111nxt − 6.6812rrpt Invest

ξ3,t+1 = yt − 0.8521lt − 0.2636kt + 0.2315rrpt − 0.1859gt + 0.0007t Product

ξ4,t+1 = yt − 0.0354kt − 0.7782ct − 0.0493nxt − 0.1070rrpt − 0.1618gt IS

ξ5,t+1 = ct − 0.4530lt − 0.4530ωt − 0.4065kt − 0.2231Rt + 0.7744rrpt − 0.2366gt Consump .


Figure: 6. Reduced form errors for the long-run relations. Estimated −ξt forthe 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 ofchained (2000) dollars, 1982-2004 from the poisson model with different λvalues compared with that from the depreciation smoothing procedure.

Figure: 10. Reduced form errors for the long-run relations. Estimated −ξt forVECM 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 , ko ,Ro , rro

[-1, 1, 1.2633, -8.6959]

Dynamic OLS:

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

United States estimates for equipment and software (E&S) investment unless stated otherwise. Variables arethose included in the investment relation with the corresponding cointegration vector at the bottom. The wagecoefficient 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 -1.1U.S. non high-tech investment -0.3 to -0.5Canadian 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 arethose included in the investment relation with the corresponding cointegration vector at the bottom.

Conclusion

I According to the estimation results for α adjustmentcoefficients, the user cost of capital is found to be statisticallysignificantly adjusting to the long-run reduced form shocks onseveral cointegration relations.

I User cost variable is endogenous to the general equilibriumsystem and responses to macroeconomic shocks includingexternal and policy shocks.

I Net exports is also endogenous to the system and needs to beincluded in the investment relation for cointegration.

I Influence of domestic savings and investment on theequilibrium long-term real interest hence on the endogenoususer cost of capital in a large open economy. Large estimateduser cost elasticities estimates may be biased due to theequilibrium current account adjustments (S − I = NX ).


Conclusion

I Small user cost elasticities for the U.S. obtained in the VECMwithout net exports but with linear time trend (close to -0.4 inChirinko et al., 2007) or with R&D capital-augmentingtechnology dt (close to the wage elasticity of -0.25 inJuselius M., 2008).

I Endogeneity of the user cost that is influenced by the demandand supply sides of a large economy hence the domesticsupply of and demand for capital.

I User cost elasticity estimate of around -1.0 for CanadianVECM. In a small open economy the interest rates and capitalgoods prices are largely predetermined internationally andmuch less affected by the domestic demand for and supply ofcapital.

I Perfectly elastic supply of capital from abroad so easier toidentify 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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