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\begin{document}
\author{{\Large Karl Whelan} \\
University Collge Dublin\thanks{E-mail: karl.whelan@ucd.ie.}}


\title{\textbf{New Evidence on Balanced Growth, \\ Stochastic Trends, and Economic Fluctuations}}
\date{June 2008 \\ JEL Codes: E32, O41}
\maketitle

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\begin{abstract}
\vspace{-0.06in} \noindent The one-sector Solow-Ramsey growth model informs how most modern researchers characterize macroeconomic trends and cycles, and evidence supporting the model's balanced growth predictions is often cited.  This paper shows, however, that the inclusion of recent data leads to the balanced growth predictions being rejected.  An alternative balanced growth hypothesis---that the ratio of nominal consumption to nominal investment is stationary---is put forward, and new measures of the stochastic trends and cycles in aggregate US data are derived based on this hypothesis. The contrasting behavior of real and nominal ratios is consistent with a two-sector model of economic growth, with separate production technologies for consumption and investment and two stochastic trends underlying the long-run behavior of all macroeconomic series.  Empirical estimates of these stochastic trends are presented based on a structural VAR and the role played in the business cycle by shocks to these trends is discussed.  
\end{abstract}
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\section{Introduction}

\vspace{-0.1in} The one-sector Solow-Ramsey growth model remains the
workhorse underlying a vast amount of macroeconomic research. In 
particular, its ``balanced-growth" prediction that the ratios of real consumption,
investment, and output should all be stationary (and thus that the logs of
the level series share a common stochastic trend) has played a key role in modern
macroeconomics.  Sample averages of these ``great ratios'' are regularly used to calibrate 
the long-run properties of a wide range of theoretical macroeconomic models.  
In addition, the common trend prediction has played an important role in modern empirical
characterizations of macroeconomic trends and cycles.  To cite a few well-known examples, King, Plosser, Stock, and Watson (1991) have developed empirical methods to identify a
time series for this common trend, while Cochrane (1994) and Rotemberg and Woodford (1996) have relied on the stationarity of the ratio of real consumption to real output to provide econometric estimates of the transitory cyclical components of consumption, investment, and output.


This paper presents new evidence on the traditional balanced growth hypothesis and on
the role of stochastic trends in the determination of U.S. macroeconomic
fluctuations. The evidence underlying the one-sector growth model is
reconsidered and rejected, an alternative balanced growth hypothesis consistent with a
two-sector framework is suggested, and the implications of this framework for macroeconomic trends and fluctuations are explored using a new identification strategy.

The contents of the paper are as follows. First, it is demonstrated that the
data now point strongly against the balanced growth hypothesis. For example, King, Plosser, Stock and Watson's evidence for the hypothesis is still widely cited.  However, the sample in their study ends in 1988; using more recent data, the traditional balanced growth hypothesis turns out to be firmly rejected.
Underlying this rejection is the fact that the ratio of real investment to real
consumption has moved substantially upwards over time and now displays no evidence of
mean-reversion.

It is important to emphasize that these results have been derived using standard National Income and Product Accounts (NIPA) data.  This implies a crucial distinction between these the evidence presented here and the more informal evidence in the well-known paper by Greenwood, Hercowitz, and Krusell (1997), who argue in favor of a model in which real equipment investment rises relative to real GDP, but in which the real equipment series is defined using Robert Gordon's (1990) alternative price index. That the balanced growth property is clearly rejected for NIPA data has important implications that go well beyond those discussed by Greenwood et al because NIPA data form the basis for the vast majority of empirical work on US macroeconomics, and much of this work relies explictly or implicitly on the balanced growth hypothesis.

Second, the paper discusses the implications of the failure of the traditional balanced growth hypothesis for the measurement of stochastic trends and cycles.  It is shown that the estimated Vector Error Correction Model (VECM) implied by the traditional one-sector balanced growth hypothesis produces unintuitive trending estimates of the cyclical components of consumption, investment and output.  Because these results contradict much of the received wisdom on this issue that was outlined in previous papers such as Cochrane (1994), the paper discusses how the incorporation of new data and revisions to historical data have contributed to overturning patterns that were previously considered to be stylized facts.  

Third, an alternative ``balanced growth'' hypothesis is put forward, which
is that the ratio of \textit{nominal} investment to \textit{nominal}
consumption is stationary and evidence is presented that this hypothesis is consistent
with US macroeconomic data. This nominal-ratio balanced growth hypothesis implies that
the logs of real consumption, real investment,
and the relative price of consumption to investment can together be
characterized according to a VECM with a single cointegrating vector.  Using this alternative
VECM, new measures of the transitory components of aggregate consumption, investment, and output are developed and the implications of these alternative measures for the interpretation of recent business cycles are explored.

Fourth, a simple structural model is proposed that is compatible with the long-run properties of the data and a new structural VAR identification method is used to explore the model's implications. The differing behavior over time of the real and nominal ratios of consumption and investment are, by definition, due a decline over time in the price of investment relative to the price of consumption, a pattern that cannot occur in the one-sector growth model.  However, this pattern is consistent with a \textit{two-sector} model of economic growth in which separate technological trends in the consumption and investment sectors drive the relative price of output in the two sectors. In addition, the existence of a single cointegrating vector among consumption, investment, and their relative price implies a common trends representations in which the long-run behavior of these variables depends on two separate $I(1)$ stochastic trends.  

We use a simple two-sector model to develop a set of long-run restrictions that allow for the identification these stochastic trends as corresponding to the states of technology in the consumption- and investment-producing sectors.   Unlike a related analysis undertaken by Fisher (2006), the identification methodology adopted in this paper relies only on the model's predictions about the long-run effects of the technology on consumption, investment, and their relative prices.  By fully exploiting these predictions, the identification strategy does not need to rely on ad hoc assumptions about the covariance structure of the technology shocks.  Once estimated, the dynamic responses to the two types of technology shocks are discussed, as are the roles played by these shocks in generating
business cycle fluctuations.  It is concluded that technology shocks play a limited role in generating business cycle fluctuations in consumption, investment, and output.   

\section{Tests of the Traditional Balanced Growth Hypothesis}

\vspace{-0.1in}  The starting point for the standard neoclassical growth
model is an aggregate resource constraint of the form $
C_t+I_t=Y_t= F \left(K_t, A_t L_t \right)$ where the production function displays
diminishing marginal returns productivity with respect to capital accumulation. It is also usually assumed that a representative consumer maximizes the present discounted value of utility
from real consumption, subject to a law of motion for capital and a process
for aggregate technology. If the technology process grows at a constant
long-run average rate, then it is well known that the model's solution features
all real variables, $C,I,K,$ and $Y$, growing at the same average rate in the long-run. Thus, ratios of any of the real aggregates will be stationary stochastic processes.  

The intuition for this property of the one-sector growth model is very general
and it holds across a wide range of specifications for 
preferences and technology. Given diminishing marginal returns to capital accumulation, a once-off increase in the savings rate can only allow capital to grow faster than output for a temporary period, and a trend in the savings rate will not be optimal for any standard specification of preferences.\footnote{See, for instance the discussion of capital-output ratio dynamics in the Solow Model in Chapter 4 of DeLong (2002)} Thus, ratios involving any of the variables capital, investment, consumption and output must be stationary.

\subsection{Evidence Based on NIPA Data}
\vspace{-0.1in}  Beyond theory, the hypothesis of stationary ``great ratios'' has been held as a crucial stylized fact in macroeconomics at least as far back as the well-known
contributions of Kaldor (1957) and Kosobud and Klein (1961). More recently, King, Plosser, Stock, and Watson (1991, henceforth KPSW), using NIPA data through 1988, presented formal econometric evidence for the balanced growth hypothesis using modern time-series
methods. KPSW's results are still widely cited as important evidence for the
one-sector model's long-run restrictions.  However, Figure 1(a) shows that the traditional balanced growth predictions appear empirically invalid once we use updated
NIPA data. The figure shows the ratio of real private fixed investment to
real consumption expenditures (the same measures of investment and
consumption used in KPSW's study). That this ratio should be stable is the
essence of the balanced growth hypothesis: The model's other long-run restrictions---such as the
stationarity of the ratio of real consumption to aggregate real output or of
real investment to aggregate real output---will follow directly from
stationarity of the ratio of consumption to investment.\footnote{This statement implicitly follows the model in defining real output as an aggregate of consumption and fixed investment.  In contrast, KPSW tested for stationarity of the ratio of real consumption to real private GDP (real GDP excluding government purchases).  This differs from an aggregate of consumption and fixed investment when net exports and inventory investment are non-zero.  The preference here for the simpler output measure is due to its consistency with the pure one-sector growth model, but the results in this section are not overturned by instead looking at ratios involving private GDP.  See Section 3.4 below.}


Figure 1(a) shows that the ratio of real investment to real
consumption has moved upwards substantially over time and shows little apparent tendency for mean-reversion. Importantly, most of the increase in this
ratio occurred after 1988, the last year in KPSW's sample. Also notable is the fact that, 
despite the substantial slump in investment during the recession beginning in 2001, the cyclical low point for this ratio still exceeded the high point for all expansions prior to the 1990s, and the ratio has reversed most of this decline by the final period shown,
2005:Q4.  Table 1 shows that formal statistical evidence confirms the absence
of mean reversion suggested by the chart. Panel A reports results from three tests of the null hypothesis that the ratio series contains a unit root.  The first is the standard Augmented Dickey-Fuller $t$-test; the second is the $DF^{GLS}$ test of Elliot, Rothenberg and Stock (1996) which has superior power to the ADF test; the third is the $\bar{M} Z_\alpha^{GLS}$ test of Ng and Perron (2001) which has been shown to have excellent size and power properties.  For each test, the lag lengths for the test regressions was chosen using Ng and Perron's Modified AIC procedure.  In all three cases, the tests do not 
come close to rejecting the unit root hypothesis at
conventional levels of significance.  

The table also reports some more direct tests of the balanced
growth hypothesis. In general, the null hypothesis of no cointegration
between the log of real investment and the log of real consumption can be
rejected. However, normalizing the coefficient on real investment to be one,
one can reject the hypothesis that the coefficients on consumption in the
estimated long-run relationship is minus one, as implied by the
one-sector model's balanced growth hypothesis. Panel B's point estimate of
-0.857 is derived from the Stock-Watson dynamic least-squares estimation
methodology and the balanced growth hypothesis that this coefficient is
minus one is tested using an asymptotically valid $t$-test.\footnote{%
The exact procedure followed here is discussed on pages 608-612 of Hamilton
(1994).} Panel C's point estimate of -0.838 is derived from the maximum
likelihood systems estimation methodology of S{\o}ren Johansen (1995), while its
test of the balanced growth hypothesis is based on a Wald test statistic derived from comparison of the log-likelihoods for the constrained and unconstrained systems (this has an asymptotic $\chi^2_1$ distribution.) In both cases, the traditional balanced growth hypothesis is
rejected at significantly greater than the one percent level.


The fact that the null hypothesis of cointegration between the
logs of consumption and investment still cannot be rejected implies that,
technically, one cannot reject the idea that there still exists a
single-common-trend representation for these two series. However, it is hard
to imagine what the \textit{economic} basis for such a single trend
representation would be. For example, in an economy with a single technology process,
why would a unit shock to that process result in a one-percent increase in
investment but only a 0.86 percent increase in consumption. And why would
this long-run relationship always have to take the form $(1,-0.86)$? 
The one-sector growth model provides a powerful intuition about
the convergent forces that make it unsustainable for
the levels of consumption and investment to have different long-run trends.
This provides a theoretical case for $(1,-1)$ as a cointegrating vector that
is completely absent for the vector $(1,-0.86)$ or any other vector $(1,-b)$ where
$b$ does not equal one.\footnote{This is not to say that one cannot find empirical models based on this alternative notion of balanced
growth.  For example, Kim and Piger (2002) use the one-sector growth model to motivate a single common trend for consumption, investment, and output, and then implement this idea by equating log-consumption with the trend and allowing for non-unit weights on the trend for log-investment and log-output.}

Figure 1(b) provides an alternative way to think about balanced
growth and long-run relationships. The figure shows the ratio of \textit{%
nominal} private fixed investment to nominal consumption. Unlike the ratio
of the real series, this ratio exhibits no apparent
trend over time. Table 2 confirms that this alternative notion of balanced
growth---that the ratio of nominal expenditures on the two categories is
stable over time---receives strong empirical support from the same
statistical tests that rejected the traditional formulation of balanced
growth. Panel A reports that the hypothesis that the nominal ratio contains
a unit root is firmly rejected by each of our tests, while Panels B and C show that point
estimates of the cointegrating vectors for the logs of the nominal series
are very close to $(1,-1)$, and statistical tests cannot come close to
rejecting this null hypothesis.

A first observation about this alternative formulation of
balanced growth is that the difference between the series in Figures 1(a) and 1(b)
is in itself evidence against the one-sector growth model. The model's
assumption that consumption and investment goods are produced using the same
technology implies that any decentralized market equilibrium must feature
them having the same price, and so real ratios and nominal ratios should be
the same thing. The differences between the two charts reflect different
price developments for the types of goods: Specifically, it reflects a
substantial decline over time in the ratio of investment prices to
consumption prices.  This relative price movement likely reflects the existence
of different production technologies for consumption and investment.


Once one allows for the idea that consumption and investment can be produced
using different technologies, and thus that their prices do not always move together, then the stability of the nominal ratio of investment to consumption also provides an intuitive
version of the same ``sustainability'' idea that underlies the 
one-sector model's balanced growth prediction.  In the one-sector context, 
the ratio of real investment to real consumption cannot trend upwards over time 
because households will not wish to allocate ever higher-fractions of their incomes towards
saving.  However, in a two-sector
model with faster technological progress in the production of investment goods, then 
real investment can grow faster than real consumption without requiring ever-increasing
sacrifices on the part of households.  Moreover, the original intuition behind the 
one-sector balanced growth hypothesis still holds as an explanation for why the nominal ratio of investment to consumption should be stationary: Increases in the household savings rate (the fraction of nominal income that is saved) will only provide a temporary boost to the growth rates of capital and output.


\subsection{Comparison with Greenwood, Hercowitz, Krusell (1997) Evidence}

\vspace{-0.1in} Before moving on to describe the implications of these
results, it is worth discussing why the focus of this paper has been restricted
to official NIPA series. In previous work
aimed at replacing the one-sector approach with a model containing multiple
technology shocks, Greenwood, Hercowitz, and Krusell (1997) cited Robert
Gordon's (1990) evidence that price inflation for durable goods is
overstated in the NIPAs because valuable quality improvements are often
ignored. Their analysis replaced the official price deflator for equipment
investment with the alternative index developed by Gordon, but used the NIPA
price deflators for all other categories.

This approach does not fit well with recent work on price
mis-measurement.  For instance, much of the recent evidence on
price mis-measurement, including some of the work of the Boskin Commission,
has focused on prices for non-manufacturing output.\footnote{%
See Boskin et al (1998).} While durable goods are one of the areas where
some of the adjustments for quality suggested by Gordon have actually been
adopted by the NIPAs (most notably for computers), there are serious
questions about the adequacy of the output deflators for sectors such as
financial services and construction. Measured productivity growth in these
sectors has been extremely weak and seems likely to have been
underestimated. Given that the nominal output series are generally
considered to be relatively accurate, these considerations suggest that
price inflation in these non-manufacturing industries may be overstated.%
\footnote{See Corrado and Slifman (1999) and Gullickson and Harper (1999).} In the
absence of convincing evidence pointing one way or the other for adjusting
the relative price of durable goods, we have chosen not to apply any such
adjustment.



\section{Implications for Measurement of Trends and Cycles}

\vspace{-0.1in}  The empirical properties just documented suggest that the appropriate theoretical framework for thinking about long-run restrictions on US macro data is one with two types of production technology (thus allowing investment and consumption prices to be different) and with balanced growth formulated in terms of stability of the nominal ratio. We will
outline such a model, and how to identify the two technology shocks and
their dynamic effects, in Sections 4 and 5.   

First, however, we will present some implications of the competing balanced growth hypotheses that hold independent of any specific theoretical framework used to derive them.   Specifically, the long-run cointegrating restrictions implied by these hypotheses provide a natural way to decompose macroeconomic time series into a stochastic trend component on the one hand, and a transitory cyclical component on the other.  We show here that the traditional one-sector balanced growth hypothesis produces unsatisfactory estimates of the cyclical components in consumption, investment, and output, and these estimates are compared with those based on the alternative nominal ratio hypothesis.  

\subsection{Traditional Balanced Growth Method}

\vspace{-0.1in} According to the Granger representation theorem, the traditional formulation of the balanced growth hypothesis---that log real investment $i_t$ and log real consumption, $c_t$ are cointegrated with a (1,-1) cointegrating vector---implies the existence of a VECM representation of the following type:
\be
\left( 
\begin{array}{c}
\Delta c_t  \\
\Delta i_t
\end{array}
\right) 
= \left( 
\begin{array}{c}
\alpha_c  \\
\alpha_i
\end{array}
\right)
+ 
B(L) 
\left( 
\begin{array}{c}
\Delta c_{t-1}  \\
\Delta i_{t-1}
\end{array}
\right)
+
\left( 
\begin{array}{c}
\gamma_i  \\
\gamma_c
\end{array}
\right) 
\left(i_{t-1} - c_{t-1} \right)
+ \epsilon_t.
\ee
The results from the estimation of a two-lag version of this VECM are reported in Table 3.\footnote{This two-lag version is consistent with results from lag-length selection tests which favor three lags in the levels specification.} The coefficients on the error-correction term (the investment-consumption ratio) are both negative, with the coefficient in the investment regression being larger.  This implies that a high value of the real investment to real consumption ratio is a negative signal for future growth in both consumption and investment, and that the ratio should tend to decline by a process of investment falling by more than consumption.  However, while these coefficient estimates technically imply a stable VAR, it should be kept in mind that both cointegration tests and unit root tests on the investment-consumption ratio reject the hypothesis that this is a satisfactory model of investment and consumption dynamics. 


This VECM can be used to derive a Beveridge-Nelson-style decomposition of consumption and investment into their stochastic trend and transitory cycle components.  Letting  $\omega$ and $\psi$ represent the unconditional expectations for $\Delta c_t$ and $i_t-c_t$, the terms in the VECM can be re-arranged to arrive at a VAR representation in  $\Delta c_t - \omega$ and $i_t - c_t - \psi$:
\be
\left(I - C(L) \right)
\left(
\begin{array}{c}
\Delta c_t - \omega \\
i_t - c_t - \psi
\end{array}
\right)
= \epsilon_t.
\ee
This can be written in companion matrix form as
\be
Z_t = A Z_{t-1} + \epsilon_t. \label{comp}
\ee
Our definition of the transitory cyclical component of consumption is
\be
c^{cyc}_t = \lim_{k \rightarrow \infty} E_t \left( c_t - c_{t+k} + k \omega \right).
\label{bevnel}
\ee
In other words, the transitory component is the expected cumulative shortfall in future consumption growth relative to its trend growth rate, so that when this component is positive, one should expect the average future growth rate of consumption to be below its trend rate.  
This can be estimated from the VAR as
\be
c^{cyc}_t  =  - e'_1 \left(A + A^2 + A^3 + .... \right) Z_t 
 =  - e'_1 A \left(I-A \right)^{-1} Z_t
\ee
where $e_1$ is a $(1,0)$ vector.
The transitory cyclical component of investment can also be estimated from this VAR as
\begin{eqnarray}
i^{cyc}_t & = &  \lim_{k \rightarrow \infty} E_t \left( i_t - i_{t+k} + k \omega \right) \nonumber \\
          & = & \lim_{k \rightarrow \infty} E_t \left( i_t - c_t + c_t - c_{t+k} + c_{t+k} - i_{t+k} + k \omega \right) \nonumber \\
          & = & c^{cyc}_t + \left(i_t - c_t - \psi \right) 
\end{eqnarray}
Figure 2(a) shows the transitory components of investment and consumption generated by the VECM reported in Table 3. Both series display the unsatisfactory property of appearing to trend upwards over time, with the values early in the sample tending to be negative and the values later in the sample tending to be positive.\footnote{It should be noted that the general pattern reported here turns out to be robust to the addition of a wide range of other cyclical variables to the forecasting VAR.}  This feature is more evident for investment than for consumption: The non-stationarity is due to both series placing some weight on the real investment-consumption ratio, and the VECM coefficients suggest that this is a more important signal for future investment growth than for future consumption growth.   

\newpage

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\newpage
\renewcommand{\baselinestretch}{1.0}

\setcounter{table}{0}

\begin{table}[tbp]
\caption{Tests of the Traditional Balanced Growth Hypothesis}
\begin{center}
\par
\begin{tabular}{lrr}
&  &  \\ \hline\hline
&  &  \\ 
\multicolumn{3}{c}{\emph{A. Unit Root Tests for $\frac{i}{c}$}}
\\ 
 &  &  \\ 
Test & \multicolumn{1}{c}{Statistic} & \multicolumn{1}{c}{5\%/10\% Values} \\
ADF $t$-statistic & \multicolumn{1}{c}{-1.59} & \multicolumn{1}{c}{-2.88/-2.57} \\ 
Elliot-Rothenberg-Stock $DF^{GLS}$ & \multicolumn{1}{c}{-1.03} & \multicolumn{1}{c}{-1.95/-1.62} \\
Ng-Perron $\bar{M} Z_\alpha^{GLS}$ & \multicolumn{1}{c}{-0.86} & \multicolumn{1}{c}{-8.10/-5.70} \\
&  &  \\ \hline
&  &  \\ 
\multicolumn{3}{c}{\emph{B. Estimated Cointegrating Vector: Stock-Watson DOLS%
}} \\ 
&  &  \\ 
Variable & Null Hypothesis & Estimates \\ 
$\log \left( i \right) $ & 1 & 1 \\ 
$\log \left( c \right) $ & -1 & -0.857 \\ 
&  &  \\ 
\multicolumn{3}{l}{$t$-test of Balanced Growth Restriction: $t=-5.16$
($p$=0.0000007)} \\ 
&  &  \\ \hline
&  &  \\ 
\multicolumn{3}{c}{\emph{C. Estimated Cointegrating Vector: Maximum
Likelihood}} \\ 
&  &  \\ 
Variable & Null Hypothesis & Estimates \\ 
$\log \left( i \right) $ & 1 & 1 \\ 
$\log \left( c \right) $ & -1 & -0.838 \\ 
&  &  \\ 
\multicolumn{3}{l}{Wald Test of Balanced Growth Restriction: $\chi^2_1=17.54$
($p$=0.00003)} \\ 
&  &  \\ \hline\hline
\end{tabular}
\end{center}
\par
{\small {\emph{Notes}: $i \equiv$ Real private fixed investment at 2000
dollars, $c \equiv$ Real consumption expenditures at 2000 dollars.
Sample is 1949:1 to 2005:4. DOLS estimation was based on four 
first-difference leads and four lags; $t$-test calculated
as in Hamilton (1994) pages 608-612. Lag length for ML VAR was three
(chosen by AIC).  See text for additional details.} }
\end{table}


\begin{table}[tbp]
\caption{Tests of the Nominal-Ratio Balanced Growth Hypothesis}
\begin{center}
\par
\begin{tabular}{lrr}
&  &  \\ \hline\hline
&  &  \\ 
\multicolumn{3}{c}{\emph{A. Unit Root Tests for $\frac{i^*}{c^*}$}}
\\ 
 &  &  \\ 
Test & \multicolumn{1}{c}{Statistic} & \multicolumn{1}{c}{5\%/10\% Values} \\
ADF $t$-statistic & \multicolumn{1}{c}{-3.21} & \multicolumn{1}{c}{-2.88/-2.57} \\
Elliot-Rothenberg-Stock $DF^{GLS}$ & \multicolumn{1}{c}{-2.94} & \multicolumn{1}{c}{-1.95/-1.62} \\
Ng-Perron $\bar{M} Z_\alpha^{GLS}$ & \multicolumn{1}{c}{-9.35} & \multicolumn{1}{c}{-8.10/-5.70} \\
&  &  \\ \hline
&  &  \\ 
\multicolumn{3}{c}{\emph{B. Estimated Cointegrating Vector: Stock-Watson DOLS%
}} \\ 
&  &  \\ 
Variable & Null Hypothesis & Estimates \\ 
$\log \left( i^* \right) $ & 1 & 1 \\ 
$\log \left( c^* \right) $ & -1 & -1.003 \\ 
&  &  \\ 
\multicolumn{3}{l}{$t$-test of Balanced Growth Restriction: $t=0.106$
($p$=0.99)} \\ 
&  &  \\ \hline
&  &  \\ 
\multicolumn{3}{c}{\emph{C. Estimated Cointegrating Vector: Maximum
Likelihood}} \\ 
&  &  \\ 
Variable & Null Hypothesis & Estimates \\ 
$\log \left( i^* \right) $ & 1 & 1 \\ 
$\log \left( c^* \right) $ & -1 & -1.007 \\ 
&  &  \\ 
\multicolumn{3}{l}{Wald Test of Balanced Growth Restriction: $\chi^2_1=0.24$
($p$=0.62)} \\ 
&  &  \\ \hline\hline
\end{tabular}
\end{center}
\par
{\small {\emph{Notes}: $i^* \equiv$ Nominal private fixed investment, $c^* \equiv
$ Nominal consumption expenditures. Sample is 1949:1 to 2005:4. DOLS estimation was based on four first-difference leads and four lags; $t$-test calculated
as in Hamilton (1994) pages 608-612. Lag length for ML VAR was five
(chosen by AIC).  See text for additional details.} }
\end{table}

\newpage


\renewcommand{\baselinestretch}{1.25}
\begin{table}[tbp]
\caption{VECM Implied by Traditional Balanced Growth Hypothesis}
\begin{center}
\begin{tabular}{lr@{.}lr@{.}l}
\\ 
\hline\hline
 & \multicolumn{2}{c}{$\Delta c_t$} & \multicolumn{2}{c}{$\Delta i_t$} \\ 
\hline \\
Constant           & -0 & 008  & -0  & 045  \\
                   & (-0& 097) & (-1 & 989) \\
\\
$\Delta c_{t-1}$   & -0 & 103  &  0  & 277  \\
                   & (-1& 275) & (1 & 245) \\
\\
$\Delta c_{t-2}$   &  0 & 258  &  0  & 367  \\
                   & (3 & 222) & (1 & 659) \\
\\
$\Delta i_{t-1}$   &  0 & 065  &  0  & 424  \\
                   & (2 & 197) & (5 & 194) \\
\\
$\Delta i_{t-2}$   & -0 & 014  &  -0 & 017  \\
                   &(-0 & 484) & (-0 & 226) \\
\\
$i_{t-1} - c_{t-1}$ &  -0 & 007  &  -0  & 031  \\
                   & (-1 & 484) & (2 & 014) \\
\\
\hline \hline
\end{tabular}
\end{center}
\par
{\small {\emph{Notes}: $i \equiv$ Real private fixed investment, $c \equiv
$ Real consumption expenditures. Sample is 1949:1 to 2005:4. $t-$statistics
are given in parentheses below coefficient estimates} }
\end{table}


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\end{document}
