forecasting the recent behavior of us business fixed investment spending: an analysis of competing...

Copyright © 2007 John Wiley & Sons, Ltd.

Forecasting the Recent Behavior of USBusiness Fixed Investment Spending: AnAnalysis of Competing Models†

DAVID E. RAPACH1 AND MARK E. WOHAR2*1 Department of Economics, Saint Louis University, Saint Louis,Missouri, USA2 Department of Economics, University of Nebraska at Omaha,Omaha, Nebraska, USA

ABSTRACTWe evaluate forecasting models of US business fixed investment spendinggrowth over the recent 1995:1–2004:2 out-of-sample period. The forecastingmodels are based on the conventional Accelerator, Neoclassical, Average Q,and Cash-Flow models of investment spending, as well as real stock prices andexcess stock return predictors. The real stock price model typically generatesthe most accurate forecasts, and forecast-encompassing tests indicate that thismodel contains most of the information useful for forecasting investmentspending growth relative to the other models at longer horizons. In a robust-ness check, we also evaluate the forecasting performance of the models overtwo alternative out-of-sample periods: 1975:1–1984:4 and 1985:1–1994:4. Anumber of different models produce the most accurate forecasts over thesealternative out-of-sample periods, indicating that while the real stock pricemodel appears particularly useful for forecasting the recent behavior of invest-ment spending growth, it may not continue to perform well in future periods.Copyright © 2007 John Wiley & Sons, Ltd.

key words business fixed investment spending; out-of-sample forecasts;mean squared forecast error; forecast encompassing

INTRODUCTION

Empirical models of US business fixed investment spending have a long tradition. This is not sur-prising, given the crucial role of investment spending in determining both long-term growth andfluctuations in aggregate activity at business cycle horizons. The empirical literature has considereda number of different models of investment spending. Among the most popular models are the Accelerator (Clark, 1917; Chenery, 1952; Koyck, 1954), Neoclassical (Jorgenson, 1963; Hall and

Journal of ForecastingJ. Forecast. 26, 33–51 (2007)Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/for.1010

* Correspondence to: Mark E. Wohar, Department of Economics, University of Nebraska at Omaha, RH-512K, Omaha, NE68182-0286, USA. E-mail: [email protected]† This is a significantly revised version of our previous paper, ‘Forecasting US Business Fixed Investment Spending’. Theresults reported in this paper were generated using GAUSS 6.0. The GAUSS programs are available athttp://pages.slu.edu/faculty/rapachde/Research.htm.

34 D. E. Rapach and M. E. Wohar

Copyright © 2007 John Wiley & Sons, Ltd. J. Forecast. 26: 33–51 (2007)DOI: 10.1002/for

Jorgenson, 1967; Jorgenson, 1971), Tobin’s Q (Tobin, 1969), and Cash-Flow (Meyer and Kuh, 1957;Duesenberry, 1958; Grunfeld, 1960).1 A common approach in the empirical literature is to includeone or more of these conventional models in a ‘horse race’ designed to identify the model (or models)that best explains US business fixed investment spending over a particular period. A partial list ofempirical studies that run horse races includes Jorgenson and Siebert (1968), Jorgenson et al.(1970a, b), Bischoff (1971), Clark (1979), Bernanke et al. (1988), Barro (1990), Blanchard et al.(1993), Oliner et al. (1995), Kopcke and Bauman (2001), and Tevlin and Whelan (2003). Many ofthese papers feature horse races comparing the out-of-sample forecasting ability of competingmodels, owing to the widely held belief that tests of out-of-sample predictive power are the moststringent tests of a model’s reliability. Furthermore, given the interest of policymakers in forecast-ing US business fixed investment spending,2 out-of-sample tests constitute a relevant test design forthe examination of forecasting models.

Out-of-sample horse races have been run over different periods in the extant literature, rangingfrom the 1960s to the mid 1990s. In the present paper, we run out-of-sample horse races involvinga number of forecasting models of US business fixed investment spending over the recent1995:1–2004:2 out-of-sample period. This period witnessed an investment ‘boom’ that was at thecenter of the longest economic expansion in US history, as well as an investment ‘bust’ that con-tributed significantly to the economic recession of 2001. It is thus interesting to compare forecast-ing models of US business fixed investment spending over this recent volatile period, and the presentpaper represents an important update of the horse race literature on US business fixed investmentspending.

We consider six forecasting models of US fixed private non-residential investment spendinggrowth at forecast horizons of 1–4, 6, and 8 quarters, with each model utilizing a different explana-tory variable (or variables). Each of the first four forecasting models uses a variable suggested byone of the conventional models cited in the opening paragraph: real business output (Accelerator);real business output divided by the real user cost of capital (Neoclassical); the market value of capitalrelative to its replacement cost or average Q (Tobin’s Q); real profits (Cash-Flow). The fifth fore-casting model follows Barro (1990) and uses real stock prices in place of average Q, as Barro (1990)argues that stock prices are a potentially better measure of the market’s assessment of the futureprofitability of investment projects than average Q. The sixth forecasting model of US fixed privatenon-residential investment spending growth is motivated by the recent work of Lettau and Ludvig-son (2002). Building on the modern Q theory of investment spending (see Abel, 1982, and Abel andBlanchard, 1986), Lettau and Ludvigson (2002) argue that variables demonstrating predictive abilitywith respect to the equity risk premium (excess stock returns) should also have predictive abilitywith respect to business fixed investment spending. The sixth forecasting model includes three ofthe excess stock return predictors—the term spread, default spread, and relative short-term interestrate—that Lettau and Ludvigson (2002) find to have in-sample predictive ability with respect to USbusiness fixed investment spending growth.3

We use a number of different econometric procedures to evaluate the forecasts generated by the different models over the 1995:1–2004:2 out-of-sample period. We first measure the accuracy of the

1 See Kopcke and Bauman (2001) and Chirinko (1993) for useful surveys of models of business fixed investment spending.2 See Bernanke (2003) and Poole (2003) with respect to monetary policy.3 Lettau and Ludvigson (2002) also consider the dividend yield and the consumption-wealth ratio (Lettau and Ludvigson,2001). However, they find that the dividend yield is not a reliable predictor of excess stock returns over the 1990s and thatthe consumption-wealth ratio only predicts investment spending at horizons beyond eight quarters.

Forecasting US Business Fixed Investment Spending 35


forecasts generated by each model using the familiar mean squared error (MSE) metric. We conductpair-wise tests for significant differences in the forecast mean squared error (MSE) across models byapplying the popular Diebold and Mariano (1995) statistic (with the Harvey et al., 1997 modification).In addition to testing for equal forecast MSE across models, we conduct pair-wise tests of forecastencompassing using the Harvey et al. (1998) statistic. Forecast encompassing allows us to examinewhether the forecasts generated by a given model contain information useful for forecasting US busi-ness fixed investment spending growth beyond the information already contained in the forecasts gen-erated by a competing model. We also test whether the forecasts generated by a given model jointlyencompass the forecasts generated by all of the other competing models using the multiple forecastencompassing test of Harvey and Newbold (2000). By analyzing the forecasts generated by compet-ing models using different econometric procedures, we should have a more complete picture of theforecasting performance of the different models over the 1995:1–2004:2 out-of-sample period.

The rest of the paper is organized as follows. The next section discusses econometric issues relat-ing to the specification of the forecasting models and outlines the procedures we use to evaluate theforecasts generated by the different models. The third section reports the empirical results, includ-ing robustness checks. The final section summarizes our main findings.

ECONOMETRIC METHODOLOGY

Specification of the forecasting modelsEarly studies, such as Jorgenson et al. (1970a, b), typically estimate conventional models of invest-ment spending with the level of real business fixed investment spending serving as the dependentvariable. A number of other studies, such as Oliner et al. (1995) and Tevlin and Whelan (2003), esti-mate conventional models with the ratio of the level of real business fixed investment spending tothe lagged capital stock serving as the dependent variable. Still other studies, such as Clark (1979)and Bernanke et al. (1988), use the ratio of the level of real business fixed investment spending toreal potential output as the dependent variable. From the perspective of modern time-series econo-metrics, the use of these dependent variables is potentially problematic, as unit root tests clearly indi-cate that the levels (or log-levels) of these variables are non-stationary.4 Importantly, the use ofnon-stationary dependent variables calls into question the reliability of the inferences made in exist-ing studies, as the standard asymptotic results on which inferences are based typically do not holdin the presence of non-stationary variables.5 It also provides an explanation for the considerable serialcorrelation often detected in models of investment spending in the extant literature; see, for example,Bernanke et al. (1988).

In an effort to specify forecasting models with a stationary dependent variable, we follow Barro(1990) and use the first differences of the log-levels (approximately equal to the growth rate) of real

4 Using data from the Bureau of Economic Analysis (BEA) and Tevlin and Whelan (2003), we tested for unit roots in thelevels (and log-levels) of US real business fixed investment spending, the ratio of real business fixed investment spendingto the lagged capital stock, and the ratio of real business fixed investment spending to real business output. Using the unitroot tests of Ng and Perron (2001) with good size and power, we cannot reject the unit root null hypothesis for these variables. The complete unit root test results, as well as all other unreported results, are available athttp://pages.slu.edu/faculty/rapachde/Research.htm.5 Of course, the relationship among the levels of non-stationary variables contains important information if the variables arecointegrated. As we discuss below, we fail to find evidence of cointegrating relationships among the levels of the variablesthat appear in the models of US business fixed investment spending we consider in the present paper.



US business fixed investment spending as the dependent variable in our forecasting models. Wefurther follow Barro (1990) by adopting an autoregressive distributed lag (ARDL) structure for ourforecasting models. This structure has also been used recently by Stock and Watson (2003) in theirextensive analysis of the ability of a host of variables to forecast real output growth and inflation inthe G7 countries. Letting ∆it = it − it−1, where it is the log-level of real fixed private non-residentialinvestment spending at time t, and , our forecasting models take the following form:

(1)

where ∆xm,t is a variable characterizing a particular investment model m (m = 1, . . . , M), h is theforecast horizon, and em,t+h is a disturbance term. The Ng and Perron (2001) unit root tests with goodsize and power clearly indicate that it ∼ I(1), so that ∆it, yt+h ∼ I(0) in equation (1). We also use theNg and Perron (2001) unit root test results to specify each of the xm,t variables that characterize thedifferent investment models such that xm,t ∼ I(1) (∆xm,t ∼ I(0)). This ensures that equation (1) con-tains only stationary variables. The first five forecasting models are defined as follows:

• ‘Accelerator’ model: x1,t ≡ log-level of real business output• ‘Neoclassical’ model: x2,t ≡ log-level of the ratio of real business output to the real user cost of

capital• ‘Average Q’: x3,t ≡ log-level of the ratio of the market value of capital to its replacement cost• ‘Cash-Flow’: x4,t ≡ log-level of real profits• ‘Stock Price’: x5,t ≡ log-level of real stock prices6

We also tested for cointegration between it and each of the xm,t variables, as it would be appro-priate to include an error-correction term in equation (1) if there is a stable long-run relationshipbetween it and xm,t. We found no evidence of cointegration, so we do not include an error-correctionterm in any of the forecasting models.7

Our sixth forecasting model takes a slightly different form from equation (1). This model employsthree of the excess stock return predictors considered by Lettau and Ludvigson (2002). Followingthe specification in Lettau and Ludvigson (2002), we include a single lag of each variable in theforecasting model, so that the ‘Return Predictors’ model takes the form

(2)

where rrelt is the relative short-term interest rate (the difference between the 3-month Treasury billyield and a 1-year backward-looking moving average), termt is the term spread (the differencebetween long- and short-term government bond yields), and deft is the default spread (the differencebetween low- and high-grade corporate bond yields). We analyze the ability of each of the six (non-nested) models to forecast US fixed private non-residential investment spending growth over the1995:1–2004:2 out-of-sample period.

y it h t t t t t h+ += + + + + +a b g g g e6 6 0 6 1 6 2 6 3 6, , , , ,D rrel term def

y i xt h m m j t j

j

q

m j m t j

j

q

m t h

m m

+ -=

-

-=

-

+= + + +Â Âa b g e, , , ,

, ,

D D0

1

0

11 2

y it h jh

t j+ = += S D1

6 The sources and construction of the data are described in the Data Appendix.7 Note that even if a cointegrating relationship actually exists, including the error correction term in the forecasting modeltypically only improves forecasting performance at longer horizons (Engle and Yoo, 1987; Clements and Hendry, 1995;Hoffman and Rasche, 1996).



In order to form out-of-sample forecasts for a given model m, we use the following recursivescheme that (apart from data availability and revisions) simulates the situation of a forecaster in realtime. We first divide the total sample of T observations into in-sample and out-of-sample portions,where the in-sample portion spans the first R observations and the out-of-sample portion the last T − R observations. The first out-of-sample forecast for model m is generated in the followingmanner. We estimate equation (1) via OLS using data available through period R.8 Using the OLSparameter estimates and the observations for ∆iR−j and ∆xm,R−j, we construct a forecast for yR+h basedon model m using the fitted equation

where are the OLS estimates of am, bm,j, and gm,j, respectively, in equation

(1) using data available through period R.9 Denote the forecast error by . Inorder to generate a forecast for yR+1+h based on model m, we update the above procedure one periodby using data available through period R + 1. We repeat this process through the end of the avail-able sample, leaving us with a set of T − R − h + 1 recursive out-of-sample forecasts correspondingto model m, .

Forecast evaluationWe compute the MSE, the most popular measure of forecast accuracy. The MSE corresponding tothe forecasts at horizon h generated by model m is defined as

(3)

We use the procedure in Diebold and Mariano (1995) and West (1996) to test for significant dif-ferences in forecasting ability between a pair of competing models. More specifically, we test H0 :MSEi,h = MSEj,h against H1 : MSEi,h ≠ MSEj,h for a pair of competing models i and j. The test statis-tic is based on the loss differential, (t = R, . . . , T − h), and takes the form

(4)

where andnh = T − R − h + 1. West (1996) shows that the DMh statistic is distributed asymptotically standardnormal when comparing forecasts from non-nested models (as we do).10 In order to improve thefinite-sample performance of the DMh statistic, Harvey et al. (1997) recommend using a modifiedDMh statistic:

d n d V d n n d d d dh h t RT h

t h h h kh

k k h t R kT h

t h h t k h h= ( ) ( ) = +( ) = ( ) −( ) −( )=−

+−

=−

= +−

+ − +1 2 110 1

1Σ Σ Σˆ , ˆ ˆ ˆ , ˆ ˆ ˆ ,f f f

DMh h hV d d= ( )[ ]-ˆ 1 2

ˆ ˆ ˆ, ,d u ut h i t h j t h+ + += -2 2

MSE ,m h m t h

t R

T h

T R h u= - - +( )-+

=

-

Â11 2

,

ˆ ,um t h t R

T h+ =

-{ }

ˆ ˆ, ,u y ym R h R h m R h+ + += -ˆ , ˆ , ˆ, , , , ,a b gm R m R j m R jand

ˆ ˆ ˆ ˆ, , , , , , ,, ,y i xm R h m R j

qm R j R j j

qm R j m R j

m m+ = - = -= + +- -Â Âa b g0 0

1 1 2 1D D

8 We select qm,1 and qm,2 in equation (1) using the SIC and the in-sample observations. We consider qm,1 values from 0 to 8.To ensure that xm,t appears in equation (1), we consider qm,2 values from 1 to 8.9 This procedure is modified in obvious ways to generate simulated out-of-sample forecasts for the Return Predictors modelgiven by equation (2).10 Note that the parameter uncertainty involved in estimating equation (1) and forming the out-of-sample forecasts does notaffect the asymptotic distribution of the DMh statistic when equation (1) is estimated using OLS. However, in general, param-eter uncertainty affects the asymptotic distributions of statistics used to analyze forecast performance; see West (1996), Westand McCracken (1998), McCracken (2000), and McCracken and West (2002).



(5)

and the tnh−1 distribution in place of the standard normal for inference. We use the MDMh statisticand the tnh−1 distribution to test for equal forecast accuracy in our applications in the third sectionbelow.11

Forecasts from two competing models can also be compared using the notion of forecast encom-passing.12 Consider forming an optimal composite forecast of yt+h as a convex combination of theforecasts from the pair of competing models i and j:

(6)

where 0 ≤ l ≤ 1. If l = 0, then the forecasts generated by model i are said to encompass the fore-casts generated by model j, as model j does not contribute any useful information—apart from thatalready contained in model i—to the formation of an optimal composite forecast. On the other hand,if l > 0, then the forecasts generated by model i do not encompass the forecasts generated by modelj, so that model j does contain information that is useful (beyond that already contained in model i)to the formation of an optimal composite forecast. Harvey et al. (1998) develop a statistic to test thenull hypothesis that the forecasts generated by model i encompass the forecasts generated by modelj (H0 : l = 0) against the alternative hypothesis that the model i forecasts do not encompass the modelj forecasts (H1 : l > 0). The statistic, which we denote as HLNh, takes the same form as the DMh sta-tistic in equation (7), with the exception that . As in Harvey et al. (1997),Harvey et al. (1998) suggest using a modified version of HLNh:

(7)

and the tnh−1 distribution for inference. We use the MHLNh statistic and the tnh−1 distribution to testfor forecast encompassing in our applications in the third section below.13

We also use the Harvey and Newbold (2000) procedure to test the null hypothesis that the fore-casts generated by, say, model 1 jointly encompass the forecasts generated by the remaining M − 1models. To understand the nature of the test, consider forming an optimal composite forecast involv-ing the forecasts generated by each of the individual M models:

(8)ˆ . . . ˆ ˆ . . . ˆ, , ,y y y yt h M t h t h M M t h+ + + += - - - -( ) + + +1 2 3 1 2 2l l l l l

MHLN HLNhh h

hh

n h n h h

n=

+ - + -( )ÈÎÍ

˘˚̇

-1 2 11

ˆ ˆ ˆ ˆ, , ,d u u ut h i t h j t h i t h+ + + += -( )

ˆ ˆ ˆ, ,y y yt h i t h j t h+ + += -( ) +1 l l

MDM DMhh h

hh

n h n h h

n=

+ - + -( )ÈÎÍ

˘˚̇

-1 2 11

11 We also compute the West and Cho (1995) chi-squared statistic that tests the joint null hypothesis, H0 : MSE1,h = . . . =MSEM,h.12 See Clements and Hendry (1998) for a textbook treatment of forecast encompassing.13 West (2001) shows that the parameter uncertainty inherent in forming the forecasts does affect the asymptotic distributionof the HLNh statistic, so that the HLNh statistic does not have a standard normal asymptotic distribution. However, West(2001) finds that the size distortions for the MHLNh statistic and the tnh−1 distribution are small in Monte Carlo experimentswhen the ratio of the number of out-of-sample forecasts to the number of in-sample observations (nh / R) is approximately0.25 or less and nh is less than approximately 40. In our applications in the third section below, nh / R is near 0.25 and nh isless than 40, so that parameter uncertainty is unlikely to lead to important size distortions in our use of the MHLNh statis-tic and tnh−1 distribution.



If l2 = . . . = lM = 0, then the forecasts generated by model 1 jointly encompass the remainingforecasts, and the remaining models do not contain information that is useful (beyond that alreadycontained in model 1) in the formation of an optimal composite forecast. Harvey and Newbold (2000)test the null hypothesis of multiple forecast encompassing using the MS*h statistic:

(9)

where (i = 2, . . . ,M), and is calculated using equation (14) in Harvey and Newbold (2000, p. 474). Harvey andNewbold (2000) recommend basing inference on the FM−1,nh−M+1 distribution. In Monte Carlo experiments, they find that the MS*h statistic has good size properties.14

EMPIRICAL RESULTS

We evaluate the simulated out-of-sample forecasts over the 1995:1–2004:2 period generated by thesix forecasting models of fixed private non-residential investment spending growth described above.The in-sample portion of the total sample covers 1963:1–1994:4, and we consider forecast horizonsof 1–4, 6, and 8 quarters. Forecast horizons of 1–8 quarters are relevant for business cycle dynam-ics and are thus likely to be of keen interest to policymakers. Considering forecast horizons out to8 quarters also helps to allow for lags in the investment spending process.

Table I reports the MSE for the different models. The Average Q model has the lowest MSE atthe 1-quarter horizon, while the Stock Price model has the lowest MSE at all of the other reportedhorizons. Either the Cash-Flow or Return Predictors model has the highest MSE at horizons of 1–4and 6 quarters, and the Accelerator model has the highest MSE at the 8-quarter horizon. The resultsin Table I point to the importance of stock prices in forecasting investment over the 1995:1–2004:2period.

Table II reports the MSE ratio and MDMh statistic for all pairs of forecasting models. Note thata ratio greater than (less than) unity indicates that the MSE for the forecasting model given in thefirst row (column) of the table is less than the MSE for the forecasting model given in the first column(row) of the table. We see from Table II that certain forecasting models often have an MSE that is

V̂ dh( )d d d d n d i M d u u uh h M h i h h t R

T hi t h i t h t h i t h t h= [ ]′ = ( ) =( ) = −( )=

−+ + + + +2 1 11 2, , , , , , , ,, . . . , , ˆ , . . . , , ˆ ˆ ˆ ˆΣ

MS*h h h h h hM n n M d V d d= -( ) -( ) - +( ) ¢ ( )[ ]- - -1 1 1

1 1 1ˆ

14 Their Monte Carlo experiments also suggest that the MS*h statistic has limited power under some circumstances.

Table I. Forecast mean squared errors: real fixed private non-residential investment spending growth,1995:1–2004:2 out-of-sample period

Model h = 1 h = 2 h = 3 h = 4 h = 6 h = 8

Accelerator 2.57 8.62 20.60 39.70 99.45 204.39Neoclassical 2.65 8.79 19.93 39.36 101.92 187.11Average Q 2.30 7.43 17.76 29.96 89.56 180.56Cash-Flow 2.83 8.89 22.37 41.48 103.17 180.36Stock Price 2.56 7.42 14.64 28.91 86.72 176.91Return Predictors 2.68 9.29 21.99 42.08 102.99 182.53

Note: A bold entry signifies the model with the lowest MSE.



Table II. MSE ratios and modified Diebold and Mariano (1995) statistics for tests of equal forecast MSE: realfixed private non-residential investment spending growth, 1995:1–2004:2 out-of-sample period

Model Accelerator Neoclassical Average Q Cash-Flow Stock Price

h = 1Neoclassical 0.97

(−0.18) [0.86]Average Q 1.12 1.15

(0.57) [0.58] (0.98) [0.33]Cash-Flow 0.91 0.93 0.81

(−0.62) [0.54] (−0.53) [0.60] (−1.12) [0.27]Stock Price 1.00 1.03 0.90 1.11

(0.02) [0.98] (0.22) [0.83] (−1.33) [0.19] (0.51) [0.61]Return 0.96 0.99 0.86 1.06 0.96Predictors (−0.31) [0.76] (−0.09) [0.93] (−0.79) [0.43] (0.32) [0.75] (−0.24) [0.81]


(−0.15) [0.89]Average Q 1.16 1.18

(0.44) [0.66] (0.53) [0.59]Cash-Flow 0.97 0.99 0.84

(−0.24) [0.81] (−0.13) [0.90] (−0.61) [0.55]Stock Price 1.16 1.19 1.00 1.20

(0.49) [0.63] (0.65) [0.52] (0.01) [0.99] (0.70) [0.49]Return 0.93 0.95 0.80 0.96 0.80Predictors (−0.55) [0.59] (−0.33) [0.75] (−0.60) [0.55] (−0.27) [0.79] (−0.74) [0.47]


(0.20) [0.84]Average Q 1.16 1.12

(0.34) [0.74] (0.24) [0.81]Cash-Flow 0.92 0.89 0.79

(−0.60) [0.55] (−0.50) [0.62] (−0.54) [0.59]Stock Price 1.41 1.36 1.21 1.53

(1.01) [0.32] (0.78) [0.44] (0.62) [0.54] (1.25) [0.22]Return 0.94 0.91 0.81 1.02 0.67Predictors (−0.60) [0.55] (−0.43) [0.67] (−0.43) [0.67] (0.14) [0.89] (−1.11)[0.27]


(0.05) [0.96]Average Q 1.33 1.31

(1.02) [0.32] (0.93) [0.36]Cash-Flow 0.96 0.95 0.72

(−0.34) [0.74] (−0.22) [0.83] (−1.01) [0.32]Stock Price 1.37 1.36 1.04 1.44

(1.07) [0.29] (0.93) [0.36] (0.73) [0.47] (1.09) [0.28]Return 0.94 0.94 0.71 0.99 0.67Predictors (−0.48) [0.63] (−0.29) [0.77] (−0.94) [0.36] (−0.20) [0.85] (−1.02) [0.31]



considerably lower than the MSE for competing models. For example, at the 3-quarter horizon, theAccelerator, Neoclassical, Average Q, and Cash-Flow models have an MSE that is 41%, 36%, 21%,and 53% higher, respectively, than that of the Stock Price model, and the Stock Price model has anMSE that is 33% lower than the Return Predictors model. Even though there are frequently sizabledifferences in MSE across models, none of the MDMh statistics are significant for any pair of modelsat conventional levels at forecast horizons of 1–4 and 6 quarters in Table II. There are three rejec-tions of the null hypothesis of equal MSE at the 8-quarter horizon.15

The MHLNh statistics corresponding to all model pairs are reported in Table III. The statistics cor-respond to a test of the null hypothesis that the forecasts for the model given in the first row of thetable encompass the forecasts for the model given in the first column of the table. Table III alsoreports the MS*h statistic for a test of the null hypothesis that the forecasts for the model given in thefirst row of the table jointly encompass the forecasts for the other five models. We see from TableIII that there are quite a large number of rejections of the null hypothesis of forecast encompassingat conventional significance levels, indicating that there are numerous situations where the forecastsgenerated by a particular model contain information useful for forecasting investment spendinggrowth beyond the information contained in another model. At the 1-quarter (2-quarter) horizon, 21

Table II. Continued

Model Accelerator Neoclassical Average Q Cash-Flow Stock Price


(−1.17) [0.25]Average Q 1.11 1.14

(0.52) [0.60] (0.83) [0.41]Cash-Flow 0.96 0.99 0.87

(−0.18) [0.85] (−0.05) [0.96] (−0.46) [0.65]Stock Price 1.15 1.18 1.03 1.19

(0.70) [0.49] (0.99) [0.33] (1.06) [0.30] (0.59) [0.56]Return 0.97 0.99 0.87 1.00 0.84Predictors (−0.56) [0.58] (−0.09) [0.93] (−0.64) [0.53] (0.02) [0.99] (−0.84) [0.41]


(0.26) [0.79]Average Q 1.13 1.04

(2.34*) [0.03] (0.10) [0.92]Cash-Flow 1.13 1.04 1.00

(0.54) [0.60] (0.26) [0.80] (0.00) [1.00]Stock Price 1.16 1.06 1.02 1.02

(3.85**) [0.00] (0.15) [0.88] (2.56*) [0.02] (0.08) [0.94]Return 1.12 1.03 0.99 0.99 0.97Predictors (0.62) [0.54] (0.18) [0.86] (−0.06) [0.95] (−0.13) [0.90] (−0.16) [0.88]

Note: †, *, ** indicate significance at the 10%, 5%, and 1% levels, respectively. The top figure is the ratio of the meansquared error for the model given in the first row to the mean squared error for the model given in the first column. Thefigure in parentheses is the modified Diebold and Mariano (1995) statistic corresponding to a test of the null hypothesis thatthe forecast mean squared errors from two models are equal against the two-sided alternative hypothesis that they are notequal; p-values are given in brackets (0.00 indicates less than 0.05).

15 The West and Cho (1995) chi-squared statistic used to test H0 : MSE1,h = . . . = MSEM,h (see footnote 11) is insignificant atforecast horizons of 1–4 and 6 quarters; it is significant at the 1% level at the 8-quarter horizon.



Table III. Harvey et al. (1998) statistics for tests of forecast encompassing: real fixed private non-residentialinvestment spending growth, 1995:1–2004:2 out-of-sample period

Model Accelerator Neoclassical Average Q Cash- Flow Stock Price ReturnPredictors

h = 1Accelerator 1.78* [0.04] 1.96* [0.03] 2.20* [0.02] 2.40* [0.01] 1.57† [0.06]Neoclassical 1.72* [0.05] 0.39 [0.35] 1.17 [0.12] 1.32† [0.10] 1.23 [0.11]Average Q 3.36** [0.00] 2.09* [0.02] 2.05* [0.02] 1.79* [0.04] 2.60** [0.01]Cash-Flow 1.17 [0.12] 0.56 [0.29] 0.64 [0.26] 1.39† [0.09] 0.88 [0.19]Stock Price 3.39** [0.00] 1.76* [0.04] −0.74 [0.77] 1.79** [0.04] 2.51** [0.01]Return Predictors 1.11 [0.14] 1.41† [0.08] 1.50† [0.07] 1.58† [0.06] 1.94* [0.03]MSh* 2.23† [0.07] 1.34 [0.27] 1.11 [0.37] 1.57 [0.20] 1.48 [0.22] 2.12† [0.09]h = 2Accelerator 1.71* [0.05] 1.37† [0.09] 2.04* [0.02] 1.54† [0.07] 1.36† [0.09]Neoclassical 1.52† [0.07] 0.78 [0.22] 1.70* [0.05] 0.83 [0.21] 1.02 [0.16]Average Q 2.36* [0.01] 2.10* [0.02] 1.94* [0.03] 0.73 [0.24] 2.18* [0.02]Cash-Flow 1.60† [0.06] 2.43* [0.01] 0.94 [0.18] 1.05 [0.15] 1.39† [0.09]Stock Price 3.06** [0.00] 2.56** [0.00] 0.73 [0.24] 2.37* [0.01] 2.47** [0.01]Return Predictors 1.05 [0.15] 0.60 [0.28] 1.06 [0.15] 1.07 [0.15] 1.10 [0.14]MSh* 2.97* [0.03] 2.60* [0.04] 1.14 [0.36] 2.05† [0.10] 0.99 [0.44] 2.73* [0.04]h = 3Accelerator 1.21 [0.12] 0.96 [0.17] 1.63† [0.06] 0.89 [0.19] 1.44† [0.08]Neoclassical 1.46† [0.08] 0.57 [0.29] 1.16 [0.13] 0.21 [0.42] 1.06 [0.15]Average Q 1.92* [0.03] 1.50† [0.07] 1.88* [0.03] 0.29 [0.39] 1.75* [0.04]Cash-Flow 0.61 [0.27] 0.72 [0.24] 0.91 [0.19] 0.59 [0.28] 0.61 [0.27]Stock Price 2.35* [0.01] 1.49† [0.07] 1.14 [0.13] 2.11* [0.02] 1.88* [0.03]Return Predictors 0.84 [0.20] 0.37 [0.36] 0.76 [0.23] 1.01 [0.16] 0.24 [0.41]MSh* 1.69 [0.17] 1.07 [0.39] 1.69 [0.17] 1.38 [0.26] 0.59 [0.71] 1.09 [0.39]h = 4Accelerator 1.23 [0.11] 0.18 [0.43] 1.21 [0.12] 0.33 [0.37] 1.35† [0.09]Neoclassical 0.99 [0.17] 0.01 [0.49] 0.86 [0.20] 0.03 [0.50] 0.79 [0.22]Average Q 1.81* [0.04] 2.04* [0.02] 1.55* [0.07] −0.01 [0.51] 1.32† [0.10]Cash-Flow 0.42 [0.34] 0.76 [0.23] 0.10 [0.46] 0.07 [0.47] 1.24 [0.11]Stock Price 2.17* [0.02] 2.14* [0.02] 1.60† [0.06] 1.70* [0.05] 1.52† [0.07]Return Predictors 0.51 [0.31] 0.38 [0.35] −0.10 [0.54] 1.13 [0.14] −0.07 [0.53]MSh* 1.18 [0.34] 2.20† [0.08] 2.23† [0.08] 0.75 [0.60] 0.60 [0.70] 0.90 [0.49]h = 6Accelerator 1.88* [0.03] −0.04 [0.51] 0.71 [0.24] −0.08 [0.53] 1.54† [0.07]Neoclassical 1.21 [0.12] −0.35 [0.64] 0.69 [0.25] −0.32 [0.62] 0.70 [0.25]Average Q 0.94 [0.18] 1.21 [0.12] 0.94 [0.18] −0.72 [0.76] 0.90 [0.19]Cash-Flow 0.53 [0.30] 0.83 [0.21] 0.31 [0.38] 0.29 [0.39] 0.81 [0.21]Stock Price 1.23 [0.11] 1.54† [0.07] 1.47† [0.08] 1.12 [0.14] 1.15 [0.13]Return Predictors 0.82 [0.21] 1.22 [0.11] −0.16 [0.56] 0.76 [0.23] −0.23 [0.59]MSh* 2.61* [0.05] 4.75** [0.00] 1.05 [0.41] 1.14 [0.36] 0.41 [0.84] 1.72 [0.16]h = 8Accelerator 0.08 [0.47] −2.05 [0.98] −0.25 [0.60] −3.08 [1.00] −0.27 [0.61]Neoclassical 0.58 [0.28] 0.23 [0.41] 0.30 [0.38] 0.20 [0.42] 0.76 [0.23]Average Q 2.51** [0.01] 0.42 [0.34] 0.32 [0.37] −2.65 [0.99] 0.40 [0.35]Cash-Flow 0.79 [0.22] 0.77 [0.23] 0.35 [0.36] 0.26 [0.40] 0.70 [0.24]Stock Price 4.16** [0.00] 0.48 [0.32] 2.48** [0.01] 0.41 [0.34] 0.50 [0.31]Return Predictors 0.97 [0.17] 0.87 [0.20] 0.47 [0.32] 1.13 [0.13] 0.36 [0.36]MSh* 6.49** [0.00] 2.71* [0.04] 2.77* [0.04] 2.30† [0.07] 1.56 [0.21] 5.86** [0.00]

Note: †, *, ** indicate significance at the 10%, 5%, and 1% levels, respectively. The statistics correspond to a test of thenull hypothesis that the forecasts for the model given in the first row of the table encompass the forecasts for the modelgiven in the first column of the table; p-values are given in brackets (0.00 indicates less than 0.05). MSh* is the Harvey andNewbold (2000) statistic corresponding to a test of the null hypothesis that the forecasts for the model given in the first rowof the table jointly encompasses the forecasts for the other models.



(18) of the 30 MHLNh statistics are significant at the 10% level, and there is no case where the fore-casts from a particular model are able to encompass the forecasts from each of the other five modelsin pair-wise tests at horizons of 1 and 2 quarters. We can also reject the null hypothesis of multipleforecast encompassing for the Accelerator and Return Predictors (Accelerator, Neoclassical, Cash-Flow, and Return Predictors) models at the 1-quarter (2-quarter) horizon.16

Two of the forecasting models begin to distinguish themselves at horizons of 3 and 4 quartersaccording to the encompassing tests in Table III. At the 3-quarter horizon, both the Stock Price andAverage Q models are able to forecast encompass the other five models in both pair-wise and jointtests. In addition, the Accelerator, Neoclassical, Cash-Flow, and Return Predictors models cannotforecast encompass the Average Q or Stock Price models in pair-wise tests. The Average Q and StockPrice models thus stand out as the best forecasting models at the 3-quarter horizon according to theencompassing tests, as none of the other models contain information useful for forecasting US realbusiness investment spending growth beyond the information contained in the Average Q and StockPrice models, and each of these two models contains information useful for forecasting investmentspending beyond that contained in the other four models. At the 4-quarter horizon, only the fore-casts generated by the Stock Price model encompass the forecasts generated by the remaining fivemodels in both pair-wise and joint tests. Furthermore, none of the other five models can forecastencompass the Stock Price model at the 4-quarter horizon. In terms of the forecast encompassingtests, the Stock Price model ranks as the best model at the 4-quarter horizon over the 1995:1–2004:2out-of-sample period. At horizons of 6 and 8 quarters, the Stock Price model continues to forecastencompass the other five models in both the pair-wise and joint tests.

The results in Tables II and III stand in sharp contrast with respect to statistical significance. Whataccounts for these differences? It may primarily be an issue of power. In extensive Monte Carloexperiments, McCracken (2004) and Clark and McCracken (2001, 2005) find that encompassingtests are typically considerably more powerful in detecting differences in predictive ability than theDiebold and Mariano (1995) and West (1996) statistic when comparing forecasts from nestedmodels.17 Of course, in the present paper we are comparing forecasts from non-nested models, andto our knowledge, there are no studies analyzing the relative power of the equal MSE and encom-passing tests when comparing forecasts from non-nested models. The results in Tables II and IIIsuggest that the MHLNh statistic is more powerful in detecting forecasting gains than the MDMh

statistic. While an extensive Monte Carlo analysis of the relative power of the MDMh and MHLNh

statistics is beyond the scope of the present paper, we ran the following small Monte Carlo experi-ment to provide some preliminary insights into whether the MHLNh statistic is likely to be morepowerful than the MDMh statistic in our applications.

Recall from Table I that the Cash-Flow model has the highest MSE at the 1-quarter horizon andthe Average Q model the lowest, with the Average Q model bringing about a 19% reduction in MSErelative to the Cash-Flow model (see Table II). This is a sizable and economically significant reduc-tion in MSE, although the difference in MSE is not statistically significant according to the MDMh

statistic in Table II. While the MDMh is not statistically significant, we see from Table III that the

16 The inability to reject multiple forecast encompassing for the other models at the 1-quarter and 2-quarter horizons is likelydue to the low power of these tests relative to the pair-wise encompassing tests. The large number of rejections at horizonsof 1 and 2 quarters in Table 3 indicates that more complicated forecasting models that include more than one of the xm,t vari-ables in equation (1) may offer forecasting gains. In the present paper, we focus on comparing six non-nested models ofbusiness fixed investment spending.17 See Rapach and Weber (2004) for an empirical application that uses both equal MSE and encompassing tests in the contextof nested models.



Average Q model forecasts encompass the Cash-Flow model forecasts, while the Cash-Flow modelforecasts do not encompass the Average Q forecasts, indicating that the Average Q model forecastsare superior to the Cash-Flow model forecasts. We assume the forecast errors from two competingmodels (for h = 1 and T − R − h + 1 = 38 forecast errors from each model) are generated by a bivari-ate normal distribution with mean vector and covariance matrix that match the sample counterpartsfor the Average Q and Cash-Flow model forecasts, so that the forecast errors are drawn from a pop-ulation where the first error has a 19% lower MSE than the second error (and the errors have thesame contemporaneous correlation as in the data). We make 5000 draws from this distribution, andfor every draw we compute the MDMh statistic using equation (5) and the two MHLNh statisticsusing equation (7) (to test whether the first model forecasts encompass the second and vice versa).We count the number of times the MDMh statistic rejects the null hypothesis of equal MSE, as wellas the number of times that we fail to reject the null that the first model forecasts encompass thesecond and reject the null that the second model forecasts encompass the first according to theMHLNh statistic. For this experiment, we are able to reject the null of equal MSE using the MDMh

statistic 30% of the time at the 10% significance level. We cannot reject the null hypothesis that thefirst model forecasts encompass the second, while at the same time we reject the null that the secondmodel forecasts encompass the first, using the MHLNh statistic at the 10% significance level 75%of the time. This small experiment suggests that encompassing tests can be substantially more pow-erful than the Diebold and Mariano (1995) and West (1996) test in detecting economically signifi-cant advantages in predictive ability when comparing forecasts from non-nested models. To allowfor departures from normality in the distribution of the forecast errors, we also ran the experimentusing a bivariate t-distribution with 6 degrees of freedom. In this case, we are only able to reject thenull of equal MSE 24% of the time using the MDMh statistic; in contrast, the encompassing testsindicate a significant advantage for the first model forecasts 73% of the time using the MHLNh sta-tistic. This again suggests that the encompassing tests are more powerful than the Diebold andMariano (1995) and West (1996) test and helps to explain the divergent results with respect to sta-tistical significance in Tables II and III.18 Overall, when we take the results in Tables I–III together,the Stock Price model often offers important forecasting gains relative to competing models overthe 1995:1–2004:2 out-of-sample period.19

In order to check the robustness of the forecasting results, we also compared the forecasts gener-ated by the different models of investment spending growth over two alternative out-of-sampleperiods: 1975:1–1984:4 and 1985:1–1994:4. Table IV reports the MSE for each of the forecastingmodels at horizons of 1–4, 6, and 8 quarters. For the 1975:1–1984:4 out-of-sample period, the Cash-

18 The fact that the Diebold and Mariano (1995) and West (1996) test uses a two-sided alternative hypothesis when com-paring non-nested models, while the encompassing tests use a one-side alternative, helps to account for some of the differ-ences in power.19 In a previous version of this paper, we tested each of the forecasting models against autoregressive (AR) benchmarks, andwe typically find that the forecasting models significantly outperform the AR benchmarks. Note that the Diebold and Mariano(1995) and Harvey et al. (1998) statistics do not have standard asymptotic distributions when comparing forecasts fromnested models, as is the case when comparing forecasts generated by equation (1) to forecasts generated by an AR bench-mark; see Clark and McCracken (2001, 2005) and McCracken (2004). We also computed forecasts of the structures andequipment components of US real fixed private non-residential spending growth using each of the forecasting models above,as well as a forecasting model based on the Tevlin and Whelan (2003) disaggregated model. In general, the forecastingmodels perform more poorly in forecasting the structures component of US real fixed private non-residential investmentspending growth than in Table I. With respect to forecasting the equipment component of US real fixed private non-residential investment spending growth, the results are somewhat similar to those reported in Tables I–III, but none of theforecasting models tends to stand out like the Stock Price model at horizons of 3 and 4 quarters in Table 3.



Flow model has the lowest MSE at horizons of 1 and 8 quarters, while the Average Q has the lowestMSE at horizons of 2–4 and 6 quarters. For the 1985:1–1994:4 out-of-sample period, the Accelera-tor model has the lowest MSE at horizons of 1–3 quarters, while the Return Predictors model hasthe lowest RMSE at horizons of 4, 6, and 8 quarters.20 A ‘problem’ with the results in Table IV isthat the Stock Price model, which typically generates the most accurate forecasts for the1995:1–2004:2 out-of-sample period, never produces the most accurate forecasts over the alterna-tive out-of-sample periods. To see how the results can be reversed over different out-of-sampleperiods, consider the relative forecasting performance of the Accelerator and Stock Price models.While the Stock Price model forecasts are always more accurate than the Accelerator model fore-casts over the recent 1995:1–2004:2 out-of-sample period, the Accelerator model forecasts are moreaccurate than the Stock Price model forecasts at horizons of 1–4 and 6 quarters over the1985:1–1994:4 out-of-sample period. In light of the results in Table IV, one should exercise cautionin using the Stock Price model to forecast US business fixed investment spending growth: while theStock Price model has performed relatively well over the recent 1995:1–2004:2 out-of-sampleperiod, it may not continue to perform well in future periods.

A natural explanation for the lack of consistent forecasting performance over different out-of-sample periods is structural instability. Stock and Watson (1996, 2003) present extensive evidenceof structural breaks in macroeconomic relationships. We examine the relevance of structural breaksfor models of investment spending growth by applying the Andrews (1993) supWald test to the

20 To conserve space, we do not report complete results for the MDMh and MHLNh statistics. The results are similar to thosein Tables II and III in that the MDMh statistics are rarely significant, while the MHLNh statistics are often significant. Asingle model does not tend to stand out over the alternative out-of-sample periods as much as the Stock Price model doesat longer horizons for the 1995:1–2004:2 out-of-sample period.

Table IV. Forecast mean squared errors: real fixed private non-residential investment spending growth, variousout-of-sample periods

Model h = 1 h = 2 h = 3 h = 4 h = 6 h = 8

1975:1–1984:4 out-of-sample periodAccelerator 5.88 16.64 34.71 58.03 107.86 123.83Neoclassical 8.05 24.14 52.42 87.76 155.86 155.68Average Q 6.09 15.26 32.83 56.83 106.30 115.42Cash-Flow 5.55 23.72 36.62 65.23 114.64 103.94Stock Price 6.11 19.47 41.18 60.89 111.76 132.65Return 6.70 21.67 52.82 97.02 194.49 239.68Predictors1985:1–1994:4 out-of-sample periodAccelerator 2.24 5.29 11.85 20.60 52.96 112.39Neoclassical 2.64 6.31 15.15 23.41 67.11 106.04Average Q 2.79 7.46 16.54 25.25 66.87 101.67Cash-Flow 2.76 6.88 14.96 26.74 60.84 107.17Stock Price 2.92 8.22 17.59 26.71 62.24 100.60Return 2.80 6.49 13.16 19.38 37.12 50.31Predictors

Note: A bold entry signifies the model with the lowest MSE.



ARDL model, equation (1), for each of the six different models (equation (2) for the Return Pre-dictors model) over the entire 1963:1–2004:2 period.21 We allow for multiple structural breaks byusing the sequential procedure of Bai (1997). The sequential procedure begins by calculating thesupWald test for the entire sample. If the supWald statistic is significant at the 10% level, we thencalculate the supWald statistic for each of the subsamples defined by the significant break. Weproceed in this manner until all of the subsamples defined by any significant breaks do not showsignificant evidence of a structural break. If the initial supWald statistic computed for the entiresample is insignificant, Bai (1997, p. 557) recommends applying the supWald test to each subsam-ple defined by the estimated break date, as the initial supWald statistic can have low power in thepresence of multiple breaks. Using the supWald statistic and the Bai (1997) sequential procedure,we obtain significant evidence of multiple structural breaks in each of the six models of investmentspending growth at all horizons.22 Given the extensive evidence of structural breaks in the models,it is less surprising that different models appear to provide the most accurate forecasts over differ-ent out-of-sample periods.23

CONCLUSION

In this paper, we run horse races involving a number of forecasting models of US real fixed privatenon-residential investment spending growth over the 1995:1–2004:2 out-of-sample period, a volatileperiod marked by an investment ‘boom and bust’ cycle. The forecasting models we consider arebased on the familiar Accelerator, Neoclassical, Average Q, and Cash-Flow models of investmentspending, and we also consider two forecasting models suggested by the more recent work of Barro(1990) and Lettau and Ludvigson (2002). The Average Q model produces the most accurate fore-casts at the 1-quarter horizon, while the Barro (1990) Stock Price model generates the most accu-rate forecasts at horizons beyond 1 quarter. At longer forecast horizons, forecast encompassing testsindicate that the Stock Price model contains most of the information useful for forecasting US busi-ness fixed investment spending growth for the 1995:1–2004:2 out-of-sample period relative to theother models. These results point to an important predictive role for the stock market with respectto the recent course of US business fixed investment spending growth. While stock prices appearimportant in forecasting the recent behavior of US business investment spending growth, robustnesschecks show that other variables are often more important over the 1975:1–1984:4 and1985:1–1994:4 out-of-sample periods. Furthermore, we obtain extensive in-sample evidence of mul-tiple structural breaks in all of the forecasting models. These results question whether the Stock Pricemodel will continue to produce the most accurate forecasts outside the 1995:1–2004:2 period, andthey suggest that structural instability will make it difficult in general to forecast US business fixedinvestment spending growth using conventional and several recently proposed models.

21 The supWald statistic is computed using the Newey and West (1987) HAC covariance estimator. Following Clark andMcCracken (2005), we use a lag truncation of 2h. We use 15% trimming at each end of the sample in computing the supWaldstatistic.22 To conserve space, we do not report the complete results for the structural break tests.23 It would be interesting in future research to examine whether the forecasting performance of the models could be signif-icantly improved by estimating the models using a window whose size is adjusted based on structural change test resultsalong the lines of Pesaran and Timmermann (2002).



DATA APPENDIX

This appendix describes the data used in the present paper.

Real fixed private non-residential investment spendingReal fixed private non-residential investment spending is the seasonally adjusted quantity index fromthe BEA, NIPA Table 5.3.3.

Real business outputReal business output is seasonally adjusted real gross value added by the business sector (quantityindex) from NIPA Table 1.3.3.

Real user cost of capitalWe measure the real user cost of capital following Tevlin and Whelan (2003), who use the Hall andJorgenson (1967) formula:

(A.1)

where Ct is the real user cost of capital, Rt is the real interest rate, d is the depreciation rate, Pt isthe price of capital relative to the price of business output, P

.t / Pt is an expected ‘capital gains’ term,

ITCt is the investment tax credit, tt is the marginal corporate income tax rate, and DEPt is the presentvalue of depreciation allowances. The real interest rate is the nominal Baa corporate bond yieldminus expected inflation, where expected inflation is measured as the average inflation rate of thebusiness output deflator over the previous 5 years and the business output deflator is derived fromnominal business output (NIPA Table 1.3.5) and real business output. We also add a constant ‘riskpremium’ that normalizes the real interest rate to 6.80%, which Tevlin and Whelan (2003) treat asthe average rate of return on physical capital. We use the depreciation rate for structures (0.05648)from Bernanke et al. (1988) and the depreciation rates for computer equipment (0.31) and non-com-puter equipment (0.13) from Tevlin and Whelan (2003). To calculate the depreciation rate for thefixed private non-residential capital stock, we take a weighted average of the structures, computerequipment, and non-computer equipment depreciation rates, where the weights are the average shares(over 1987–2003) of structures, computer equipment, and non-computer equipment in the capitalstock. The capital stock data are from NIPA Table 2.1. The price of capital relative to business outputis the price deflator for fixed private non-residential investment spending divided by the businessoutput price deflator. The price deflator for fixed private non-residential investment spending is con-structed from the nominal and real fixed private non-residential investment spending series. The realinvestment series is described above, and the nominal investment spending series is from NIPA Table5.3.5. The ‘capital gains’ term is measured as a 3-year moving average of the percentage change inPt. The investment tax credit and marginal corporate income tax rates are from the Federal ReserveBoard. DEPt is computed using the ‘sum of the year’s digits’ approach of Hall and Jorgenson (1967,equation (8)), where the service life is a weighed average of the service lives a number of the com-ponents making up fixed private non-residential investment spending.24 The weights are the sharesof each of the components in nominal investment.

C R P Pt t t t t t t t= + - ( )[ ] - - ◊( ) -( )[ ]d t t˙ 1 1ITC DEP

24 We thank Stacey Tevlin for kindly providing us with the investment tax credit, marginal corporate income tax rate, andservice lives data.



Average QAverage Q is calculated using the tax-adjusted formula from Bernanke et al. (1988) as described inLettau and Ludvigson (2002, p. 64):

(A.2)

where Vt is the market value of equity, Bt is the present value of the depreciation allowances still tobe taken on the existing capital stock, Dt is the value of total liabilities in non-farm, non-financialcorporate business, and At is the current-dollar value of the stock of equipment, non-residential struc-tures, and inventories for non-farm, non-financial corporate businesses. Vt, Dt, and At are from theFederal Reserve Board’s Flow of Funds Accounts for the United States. Bt is calculated using Bt = KTAXttt{dT

t /[d Tt + RBt(1 − tt)]}, where KTAXt is the nominal stock of equipment and non-

residential structures in non-farm, non-financial corporate business that has not been depreciated fortax purposes, dT

t is the rate of tax depreciation, and RBt is 10-year Treasury bond yield. To generatethe KTAXt series, we begin with the nominal capital stock from the end of 1964 (from the BEAreport, ‘Fixed Assets and Consumer Durable Goods for 1925–2001’25) and use the relation KTAXt

= KTAXt−1 − CCAt + INt, where CCAt is capital consumption allowances (NIPA Table 1.7.5) andINt is nominal fixed private non-residential investment spending. Given KTAXt, we can calculatethe rate of tax depreciation using dT

t = CCAt/KTAXt.

Real profitsReal profits are measured as seasonally adjusted nominal after-tax corporate profits with inventoryvaluation and capital consumption adjustments (NIPA Table 1.12) deflated by the business outputdeflator.

Real stock pricesReal stock prices are the S&P 500 stock price index deflated by the business output deflator. TheS&P 500 index is from Global Financial Data.

Relative short-term interest rateThe relative bill rate is the 3-month Treasury bill yield minus a 1-year backward-looking movingaverage. The 3-month Treasury bill yield is from Global Financial Data.

Term spreadThe term spread is the difference between the 10-year government bond yield and the 3-month Treasury bill yield. The 10-year government bond yield is from Global Financial Data.

Default spreadThe default spread is the difference between the corporate Baa bond yield and corporate Aaa bondyield. The corporate bond yields are from Global Financial Data.

QV B D

At

t

t t t

tt t t=

-ÊË

ˆ¯

- ++ + -Ê

Ëˆ¯

1

11

ttITC DEP

25 The report is available at http://www.bea.gov/bea/ARTICLES/2002/09September/0902FixedAssets.pdf.



ACKNOWLEDGEMENTS

The comments of Todd Clark, Allan Timmermann (Departmental Editor), and two anonymous ref-erees have helped to substantially improve the paper. The paper has also benefited from the com-ments of seminar participants at the 2004 Midwest Econometrics Group Meetings, 2004 MidwestMacro Meetings, 2004 Missouri Economics Conference, Federal Reserve Bank of Saint Louis,IUPUI, Saint Louis University, Texas A&M University, and the University of Texas at Arlington.The usual disclaimer applies.

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Authors’ biographies:David E. Rapach is an Associate Professor of Economics at Saint Louis University. His research interests includetime-series econometrics, macroeconomics, international finance, and financial economics. He has published innumerous journals, including Economic Inquiry, International Journal of Forecasting, Journal of Applied Econo-metrics, Journal of International Economics, Journal of International Money and Finance, Journal of Macro-economics, and Journal of Money, Credit, and Banking.

Mark E. Wohar is a UNO CBA Distinguished Professor of Economics and Finance at the University of Nebraskaat Omaha. His research interests include time-series econometrics, macro-economics, international finance, financial institutions, and financial economics. He has published more than 65 articles in economic and financialjournals, including the American Economic Review, International Journal of Forecasting, Journal of AppliedEconometrics, Journal of Finance, Journal of International Economics, Journal of International Money andFinance, Journal of Money, Credit, and Banking, and Review of Economics and Statistics.

forecasting the recent behavior of us business fixed investment spending: an analysis of competing...

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