Is the Feldstein-Horioka Puzzle Really a Puzzle?*
by
Daniel Levy
Department of Economics
Emory University
Atlanta, GA 30322
Ph: (404) 727-2941
Fax: (404) 727-4639
econdl@emory.edu
December, 1998
JEL Codes: F21, F41, C32
* I thank the late Martin J. Bailey, Marianne Baxter, Tamim Bayoumi, Yunhui Chen, Betty Daniel, Hashem
Dezhbakhsh, Clive Granger, Fuku Kimura, Kajal Lahiri, G.S. Maddala, Maurice Obstfeld, and seminar participants
at Emory and SUNY/Albany for useful comments and discussions, and John Sutton and the Royal Economic
Society for Research Grant Award. All errors are mine.
Is the Feldstein-Horioka Puzzle Really a Puzzle?
Abstract
The empirical evidence that national saving and domestic investment are correlated, also known as
Feldstein-Horioka (1980) Puzzle, has received significant attention in the literature since Feldstein
and Horioka interpret it as an evidence of low international capital mobility. This paper argues that
there is nothing mysterious in the finding that investment and saving are correlated. Since the
neoclassical growth theory predicts that in the steady state investment and saving will be
proportional to output and therefore will grow at the same rate, it would be surprising if we did not
find high long-run correlation between investment and saving. Using a dynamic intertemporal
optimization model of open economy, it is shown that long-run investment saving correlation
directly follows also from the dynamic budget constraint of open economy and this does not
depend on the degree of international capital mobility. Therefore, unless the dynamic budget
constraint is violated, the time series of investment and saving should be cointegrated, and this
would be true for any degree of capital mobility. Using an improved econometric technique which
encompasses the tests used by previous authors, I show that their conflicting findings can be
explained by a simple, but important, omitted variables problem. Using annual and quarterly postwar U.S. data, I find that investment and saving are cointegrated in levels as well as in ratios (as a
fraction of output) regardless of the time period covered, as predicted by the neoclassical growth
theory. This indicates that the U.S. economy is solvent in the sense that it does not violate its
dynamic budget constraint. Therefore, I conclude that the observed investment-saving correlation
cannot be useful in measuring capital mobility.
1
1.
Introduction
There is strong empirical evidence that national saving and domestic investment are
correlated.1 Since most of this evidence is based on cross-section regressions of multi-year average
data, this empirical finding has been interpreted as a long run phenomenon. The finding, also known
as the Feldstein and Horioka (1980) puzzle, has received significant attention in the literature sinceu
and Feldstein and Horioka interpret it as an evidence of low international capital mobility. Their
argument is as follows: in a closed economy all the domestic investment must be financed by domestic
saving and therefore we would expect domestic investment and national saving to be correlated.
However, in an open economy, at least part of the domestic investment could be financed by foreign
saving and therefore national saving and investment could in principle diverge and move
independently from each other. Therefore, Feldstein and Horioka (1980) argue, the finding that
investment and saving are correlated in highly open economies, for example, in the OECD countries,
suggests that capital might not be internationally mobile.
This interpretation of the investment-saving comovement is very controversial since it is in
contrast with the general deregulation of capital markets and increased integration of world financial
markets that took place over the last 20–30 years. In addition, empirical studies that measure capital
mobility directly using various PPP, interest rate differential and other equilibrium conditions,
conclude that capital is very mobile.2
It is important to have a good estimate of the true degree of capital mobility for several reasons.
First, the effect of fiscal policy (e.g., increased expenditures) on investment and external balance
depends crucially on the degree of capital mobility. Second, economy’s access to open capital
markets can reduce the cost of adjustment to negative exogenous shocks (e.g., natural disasters such
as earthquakes or hurricanes). Third, whether a tax policy, designed to directly affect investment
(saving) without directly affecting saving (investment), will improve or worsen the economy’s
international competitiveness depends on the degree of capital mobility.3 Fourth, degree of capital
1 See, e. g., Sinn (1992) and references cited therein. For an evidence on developing countries, see Fry (1986).
2 See, e. g., Bayoumi (1990), Sachs (1981), Purvis (1985), Frankel (1986, 1991), Obstfeld (1986a, 1986b), Popper
(1990), Baxter and Crucini (1993), Hutchison and Singh (1993), and Levy (1995, 1999).
3 For example, if capital is perfectly mobile, then investment taxes will be completely born by labor.
2
mobility is important determinant of the rate at which poor and rich economies converge. Fifth, the
assumption of perfect capital mobility is often employed in macroeconomic and financial (theoretical
as well as econometric) formulations. If capital is not really mobile, then it would question the
validity of this common practice.4
Various authors examine the time series properties of the investment-saving comovement and
report conflicting result. For example, using the post-war U.S. data, Miller (1988) examines the
investment-saving comovement and finds that saving and investment are cointegrated during a fixed
exchange rate regime, but not during a flexible exchange rate regime. Following Feldstein’s line of
argument, he concludes that increasing capital mobility since the beginning of 70s might have severed
the long-run link between investment and saving. Gulley (1992) re-examines the investment-saving
comovement using an improved test and finds that the levels of saving and investment are not
cointegrated in either periods. Similarly, Otto and Wirjanto (1989) conclude that national saving and
investment in the U.S. are not cointegrated.
In this paper I argue that there is nothing surprising in the finding of Feldstein and Horioka
(1980). Since the neoclassical growth theory predicts that in the steady state investment and saving
will be proportional to output, it would be surprising if we did not find high correlation between
investment and saving. Therefore, I argue that the time series of investment and saving should be
cointegrated, and this would be true for any degree of capital mobility.5 Using an improved
econometric technique which encompasses the tests used by Miller (1988), Otto and Wirjanto (1989),
and Gulley (1992), and which avoids some pitfalls associated with their tests, I show that their
conflicting findings can be explained by a simple omitted variable problem. In particular, using
annual and quarterly post-war U.S. data, I demonstrate that even if investment and saving are not
cointegrated in a bivariate setup, they are cointegrated when output is added to the system. In order to
allow for the possibility of structural breaks in the investment-saving relationship, I consider the entire
post-war sample period as well as several subperiods separately. I find that the trivariate cointegration
4 I shall mention that the focus of this paper is long-run capital mobility since short run capital mobility is less
controversial. See, for example, Feldstein and Horioka (1980). For a time series evidence on the short-run investmentsaving comovement and its implications for capital mobility see Levy (1999).
5 For example, Barro, et al. (1992) construct an open economy version of the neoclassical growth model with the
same conclusion.
3
finding is robust regardless of the time period covered.
The modern optimization-based dynamic model of open economy also predicts that investment
and saving will be correlated in the long run. To see this, consider a dynamic model of an open
economy where individuals optimize intertemporally. In such a framework the individuals face an
intertemporal budget constraint which allows for the possibility that they may incur debt during some
periods of their lives to smooth consumption. However, debt cannot be accumulated forever. The
existence of such an external budget constraint implies that in the long run the periodic current account
balances should add up to zero since current account surpluses or deficits cannot be sustained forever.
Thus, periods of current account surpluses must be followed by periods of current account deficits,
and visa versa. This implies that in the long run investment and saving would be correlated,
regardless of the degree of capital mobility, as long as the intertemporal budget constraint is not
violated. In this framework, then, a test of cointegration between investment and saving can be
viewed really as a test of country’s economic solvency. Since we find that investment and saving are
cointegrated, it follows that the U.S. data does not violate country’s intertemporal budget constraint
and so the U.S. economy is solvent. The main conclusion is that the observed long-run investmentsaving correlation cannot be useful in measuring the degree of long-term capital mobility.
The paper is organized as follows. In the next section I discuss omitted variables problem in
cointegration tests. In section 3, I present integration test results. The bivariate and trivariate
cointegration test results are reported and discussed in section 4. In section 5, I derive long-run
implications of intertemporal budget constraint that open economies face and reinterpret the empirical
findings in light of these implications. The paper ends with a brief summary and concluding remarks.
2.
Omitted Variables Problem in Cointegration Tests
Miller (1988), Otto and Wirjanto (1989), and Gulley (1992) use cointegration methodology to
study the investment-saving relationship in the post-war U.S. They use cointegration analysis since
the time series of saving and investment tend to be nonstationary, and therefore looking at the
correlation coefficient between them for determining their long run relationship is meaningless. Either
saving and investment are cointegrated, which means that the asymptotic correlation coefficient is one,
or they are not cointegrated, which means that the asymptotic correlation coefficient is zero.
4
Therefore, if investment-saving correlation is indeed a long run phenomenon as Feldstein and
Horioka (1980) suggest, then they must be cointegrated.
In their studies Miller (1988), Otto and Wirjanto (1989), and Gulley (1992) use Engle and
Granger’s (1987) two-step estimation method to examine the bivariate relationship between national
saving and domestic investment but they report conflicting empirical findings. Miller (1988) finds
that the series are cointegrated prior to 1971 during a fixed exchange rate regime, but but not after
1971 during a flexible exchange rate regime. Otto and Wirjanto (1989) and Gulley (1992), however,
find that the series are not cointegrated in either period.
The main point I want to make in this paper is that Feldstein-Horioka puzzle is not a puzzle at
all. To the contrary: we would expect investment and saving to be correlated and it would be
surprising if they were not. To see this, consider the neoclassical growth model [e.g, Solow (1970)]
which is what, I believe, most students of growth have in mind when they think of long-run. In that
model, saving determines investment and, in the steady state, both are proportional to output.
Therefore, we would expect to see a cointegration between saving and investment regardless of the
degree of capital mobility.6 Thus, the findings of Miller (1988), Otto and Wirjanto (1989), and
Gulley (1992) that investment and saving are not cointegrated is puzzling and may be due to statistical
misspecification.
In this paper I propose to resolve this problem by examining a possibility that some important
variables may be missing from the bivariate setup considered by the above authors. It is possible that
two variables are not cointegrated but they will be cointegrated if a third variable is added to the
system. As an example, consider a situation where y, x1, and x2 are all I(1) but their linear
combination is I(0).7 In other words, y, x1, and x2 are cointegrated and therefore
y = β 1 x1 + β 2 x2 + ε, where ε ~ I(0). Now suppose that we mistakenly omit x2 and only examine the
relationship, y = βx1 + µ. Since µ = β 2 x2 + ε ~ I(1), we would mistakenly conclude that y, and x1
are not cointegrated. Thus, I examine the possibility that the conflicting results reported by Miller
(1988), Otto and Wirjanto (1989), and Gulley (1992) may be caused by omission of some important
variables. According to the neoclassical growth model, a natural candidate for one such variable is
6 Virtually all other macro models, with or without open capital markets, make a similar prediction on the long-run
investment-saving comovement.
7 This example draws on Maddala (1992).
5
output since in that model, investment and saving are proportional to output, and therefore,
investment, saving, and income will be cointegrated. Before testing for cointegration, unit root
properties of individual time series need to be examined. This is done in the next section.
3.
Integration Test Results
Recall that, if a time series xt has to be differenced d times to make it stationary, then we say
that xt is integrated of order d, xt ~ I(d). The number of differencing needed to make a time series
stationary corresponds to the number of unit roots the series contains. To test for stationarity, Miller
(1988) uses the following variant of the Augmented Dickey-Fuller (ADF) unit root test,
∆ xt = γ xt – 1 +
4
Σ
i =1
φi ∆ xt – i + εt
(1)
where xt is the series we are examining. However, Gulley (1992) correctly argues that the exclusion
of a constant term from (1) is appropriate only if the mean of the series is zero, which obviously is not
the case for saving or investment. Therefore he modifies (1) by adding a constant,
∆ xt = α1 + γ xt – 1 +
4
Σ φi ∆ xt – i + ε t.
i =1
(2)
and tests for γ = 0.
Unfortunately, this version of the ADF test is not problem free either. In fact, for resolving the
problem of the lack of a constant term in (1), it is not suitable at all. The reason is that as several
authors have shown [see, for example, West (1988), Hylleberg and Mizon (1989), and MacKinnon
(1991)], the tabulated distribution of the unit root test statistic for version (2) of the test depends
crucially on the assumption that α1 = 0. That is, it has the DF distribution only when there is no drift
term in the data generating process of xt,. If the true α1 ≠ 0, then the statistic for testing the hypothesis
γ = 0 is asymptotically distributed as N(0, 1), and in finite samples its distribution may or may not be
well approximated by the DF distribution.8 Therefore, if the drift parameters in the data generating
processes of investment and saving are nonzero, then using version (2) of the test is not appropriate.
8 A similar problem arises in the estimation of cointegrating regressions using Engle-Granger two-step method, where
the residuals’ ADF unit root test statistic’s distribution depends on the true value of the intercept term. As MacKinnon
(1991) argues, all tables assume that α1 = 0, and therefore they may be quite misleading if that is not the case.
6
To avoid the dependence of the test statistic’s distribution on the value of α1, Engle and Yoo
(1991) and MacKinnon (1991) suggest to add a linear time trend to (2),
∆ xt = α0 t + α1 + γ xt – 1 +
4
Σ
i =1
φi ∆ xt – i + εt
(3)
under the assumption that there is no trend term in the data generating process.9 I use this variant of
the test to examine the unit root properties of saving, investment, and output. Otto and Wirjanto
(1989) also use this variant of the test.
The critical values I use for integration testing are generated by MacKinnon (1991), which is
the most complete tabulation of DF and ADF test statistics available, since it provides asymptotic
critical values for any number of observations for systems with up to six variables.10 I use quarterly
as well as annual data for the period 1947–1987. The quarterly seasonally adjusted data I use is the
same CITIBASE data set used by Miller (1988), Otto and Wirjanto (1989), and Gulley (1992).
Some of the studies of long-run investment-saving comovement, including the original
Feldstein and Horioka’s (1980) paper, use investment-to-output and saving-to-output ratios.
However, this practice has been criticized by Tobin (1983) and others who suggest that the proper
way to model the investment-saving relationship is by using the variables measured in levels.11 To
avoid these difficulties, I examine the investment-saving relationship in both forms, in levels as well
as in ratios.12
9 Including a trend in this equation often makes sense for a number of other reasons, as Engle and Yoo (1991) argue.
10 Older tabulations do not provide asymptotic critical values and they cover only few finite sample sizes. In addition,
MacKinnon (1991) uses response surface regressions in which critical values depend on the sample size, and his results
are based on 25,000 replications, rather than the 10,000 replications used by previous researchers.
11 The main reason for the use of ratios was to avoid the difficulties the presence of integrated variables create in
traditional regression analysis in the form of potential spurious results. However, it turns out (see section 3) that
measuring the time series of investment and saving in ratios is not sufficient to make them stationary.
12 Note that the results derived from variables measured in levels or in ratios are sufficient to make the main point
since in the framework of the cointegration methodology adopted here, a bivariate cointegration between investment-tooutput and saving-to-output ratios is equivalent to a trivariate cointegration between the levels of investment, saving,
and output. However, I chose to present the results based on both since there is no clear consensus in the literature on
which of the two forms, levels or ratios, is more appropriate. For example, Miller (1988) considers only investment
and saving ratios, Otto and Wirjanto (1989) consider only investment and saving levels, and Gulley (1992) considers
both, levels as well as ratios.
7
I apply the unit root test given by (3) to the time series of national saving (S ), domestic
investment (I ), and output (Y ). In order to allow for the possibility of structural break in the
investment-saving comovement, in addition to the entire period 1947:1–87:3, I also examine its
subperiods, 1947:1–71:2, 1971:3–87:3, and 1980:1–87:3. The 1947:1–87:3 sample period was
chosen to match the sample periods used by Miller (1988), Otto and Wirjanto (1989), and Gulley
(1992). The first two subperiods correspond to the fixed and flexible exchange rate regimes,
respectively.13 The last subperiod, 1980:1–87:3, is examined to see whether the huge capital
inflow to the U.S. during Reagan’s period may have altered the relationship between investment and
saving. The annual series measured in levels (log) and as a fraction of output are displayed in
Figures 1 and 2, respectively. Similarly, the quarterly series measured in levels (log) and as a
fraction of output are shown in Figures 3 and 4, respectively.
I present the integration test results for the annual and quarterly series in Tables 1 and 2,
respectively. Whenever possible, along with my corrected ADF t-statistics I also present the
statistics reported by Miller (1988), Otto and Wirjanto (1989), and Gulley (1992).14 Not
surprisingly the t-statistics I get for the quarterly data are different from the statistics reported by the
other authors. The difference between my t-statistics and the statistics reported by Otto and
Wirjanto (1989) is due to the differences in the sample period.15 As the corrected ADF statistics
indicate, saving, investment, and output series are nonstationary when measured in levels. When
measured in first differences, all three series appear to be stationary as the null hypothesis of unit
root is always rejected. Based on these findings I conclude that all three series are I(1) when
measured in levels and therefore they can be used in a joint cointegration analysis. This is true for
the annual (Table 1) as well as quarterly data (Table 2).
When measured in ratios, the time series of investment and saving appear to be I(1) during
1971:3–87:3 and 1980:1–87:3 subperiods.16 In other sample periods both series appear stationary
13 To conserve degrees of freedom, I do not split the annual data into subperiods because the cointegration test used here
uses a maximum likelihood procedure based on error correction representation of the VAR formed by the variables
considered here.
14 All these authors consider quarterly data only and therefore the corresponding columns in Table 1 are blank. As
mentioned above, Miller (1988) considers only ratios, Otto and Wirjanto (1989) consider only levels, and Gulley (1992)
considers both, levels as well as ratios.
15 See the footnotes underneath Table 2.
8
as the hypothesis of unit root is rejected each time. The finding that investment-to-output and
saving-to-output ratios are nonstationary may be in contrast with the prediction of the neoclassical
growth model.17 The reason might be the relatively low number of observations contained in these
two subperiods: it is a well known fact that the power of any statistical test to reject the null
hypothesis declines as the number of observations drops.18 In what follows I treat investment and
saving ratios as nonstationary. I now turn to cointegration analysis.
4.
Cointegration Test Results
For cointegration analysis I use Johansen’s (1988) maximum likelihood (ML) method. Although
alternative techniques are available, studies have shown that Johansen’s ML procedure has the best
properties in the sense that it yields the least biased and the most symmetrically distributed coefficient
estimates. This is true even when the errors are not normally distributed or when the underlying
dynamics are unknown [Gonzalo (1989)]. In particular, Johansen’s method is superior to the EngleGranger (1987) two-step method used by Miller (1988), Otto and Wirjanto (1989), and Gulley (1992).
In addition to the inferior statistical properties of its estimators, the Engle-Granger method has the
16 The reader might be puzzled by the finding that the investment and saving series are I(1) in levels as well as in ratios
at the same time (during the 1971:3–87:3 and the 1980:1–87:3 subperiods). That is, is it technically possible that two
variables, say saving and output be I(1) and their ratio, saving/output also be I(1)? Granger and Hallman (1988) show
that the answer to this question is yes. To see this, start with St = St – 1 + εt ~ I(1) and Yt = Yt – 1 + τt ~ I(1) and compute
S S + εt
1
the saving ratio, st = t = t – 1
, using Taylor expansion assuming that Yt is positive and large. Ignoring O Yt ––21
Yt Y t –1 + τ t
terms, this yields
st =
St – 1 εt τ t S t –1
+ – 2 + O Yt –3
–1
Yt –1 Yt
Yt – 1
= st – 1 +
εt τ t st – 1
–
+O Y t–3
–1
Yt
Yt – 1
= 1– m t st – 1 +
εt
– τ t – mt st –1 + O Yt–– 31
Yt
τt
1
1
and O Yt ––31 denotes the terms that are at most of order Yt–– 31 in magnitude. Thus, for mt >0 , st is
Yt – 1
a random walk with slowly changing drift, and if mt =0 , then st is approximately a random walk.
where mt =E
17 That model predicts that in the steady state investment and saving shares in output will be stationary.
18 Another reason for this finding might be the downward trend in the investment and saving ratios during the 80s.
Since investment and saving ratios are bounded from above as well as below, it is possible that if given enough time,
both series would look more stationary.
9
disadvantage that for estimating a cointegration relationship, some kind of normalization is necessary.
Practical applications have shown that the results can be very sensitive to the normalization chosen, i.e.,
the result depends on which variable is chosen as the left-hand-side variable. Although the EngleGranger test is asymptotically invariant to the normalization chosen, in finite samples the test results
may be very sensitive to it [for examples, see Banerjee at al. (1993)]. As Hendry (1986) shows, the
normalization problem is not as severe in a bivariate setup since in that case both regressions yield
1
virtually identical coefficients if R2 is close to unity, which is almost always the case with the time
series of investment and saving in the U.S. However, with three or more variables, which is the case
here, the problem becomes important. Johansen’s method has the advantage that it treats all variables as
endogenous and therefore avoids the problem of choosing normalization altogether.
The cointegration test results are presented in Tables 3–7. The critical values I use are taken
from Osterwald-Lenum (1992).19 In estimating the cointegration vectors, I set the order of the
vector autoregression to 4. Since it is not known a priori whether the true data generating process
contains a deterministic trend or not, I conduct the cointegration test under both possibilities. The
test statistics are identical under both assumptions, only the critical values differ. (See the last four
columns in Tables 3–7.)
In Johansen’s framework the number of cointegrating vectors is determined sequentially. We
start with the hypothesis that there are no cointegrating relations, that is, r = 0, where r denotes the
number of cointegrating relationships. We continue only if this hypothesis is rejected. In that case
we test the hypothesis that there is at most one cointegrating vector (r ≤ 1), and so on. The test results
can be interpreted in favor of cointegration only if 0 < r < m , where m denotes the number of
variables in the data vector xt . Full rank, that is r = m, only indicates that the vector process xt is
stationary. If r = 0, then the matrix Π, which is the matrix of coefficients on the variables xt – p in the
vector autoregression model written in first differences, is the null matrix and then the model becomes
a traditional differenced vector autoregression model.
Johansen (1988) proposes two tests for estimating the number of cointegrating vectors and both
are employed in this paper. The first, called maximal igenvalue test, is given by the statistic λmax, and
19 I choose not to use more commonly used critical values tabulated by Johansen and Juselius (1990) since Podivinsky
(1990) suggests that those critical values may be invalid for sample sizes of 100 or smaller.
10
is designed to test the hypothesis H(r – 1) against the hypothesis H(r). The second test statistic JT,
called trace test, is designed for testing the hypothesis H(r) against the hypothesis H(m), where r < m.
4.1
Bivariate System: Levels
Following the sequential procedure outlined above, we find that in the bivariate system,
investment and saving levels are cointegrated during 1947:1–87:3 (with one cointegrating vector, i.e.,
r = 1). To see this, look at the first and third rows in Table 4. The values of both test statistics,
λmax = 18.58 and JT =20.97, are significantly bigger than the corresponding critical values.
Therefore, the null of zero cointegrating vectors (r = 0) is rejected in favour of the alternative. Now
we proceed to the next hypothesis, r ≤ 1 (the second and fourth rows in Table 4). Here we see that
both test statistics (2.39 and 2.39) are below the corresponding critical values.20 Therefore the null
of one or less cointegrating vectors cannot be rejected. Since the hypothesis of zero cointegrating
vector was rejected in the preceding stage, we conclude that r = 1.
For the 1947:1–71:2, we find that (see Table 5) the null of no cointegration, i.e., r = 0, cannot
be rejected. For the 1971:3–87:3 subperiod the result is not conclusive since with-trend specification
of the test indicates one cointegrating vector while no-trend specification indicates only stationarity
(r = 2). When measured in ratios (Table 6), both test statistics indicate that investment and saving are
cointegrated with one cointegrating vector.21
For the 1980:1–87:3 period, the results support the view that investment and saving are
cointegrated: with no-trend specification, the null of one cointegration vector cannot be rejected. The
with-trend cointegration test indicates that the null of one cointegrating vector can be rejected only at
10% significance but not at 5% significance.
4.2
Bivariate System: Ratios
Here we find that for the 1971:3–87:3 (see Table 6) and 1980:1–87:3 (see Table 7) subperiods,
both test statistics uniformly reject the null of zero cointegrating vectors but cannot reject the null of
20 The equality of these statistics is not a coincidence. In the final stage of sequential testing, their values always equal
by construction of these tests.
21 Since there is a possibility that investment and saving ratios may contain no deterministic trend, I have also tested
for cointegration with the restriction, µ = 0, where µ is the vector of constants in the vector autoregression. The result
is identical, i.e, I find that the two series are cointegrated with one cointegration vector.
11
one cointegrating vector. Thus, domestic investment and national saving during these subperiods are
cointegrated.
In sum, the cointegration test results for domestic investment and national saving are somewhat
mixed (although they in general indicate cointegration when quarterly data is used).22 The findings
reported here differ significantly from the findings reported by Miller (1988), Otto and Wirjanto
(1989), and Gulley (1992): Miller (1988) finds that saving and investment are cointegrated during a
fixed exchange rate regime (1947:1–71:2), but but not during a flexible exchange rate regime
(1971:3–87:3), so, Miller’s findings are opposite to what we find here. Otto and Wirjanto (1989) and
Gulley (1992) conclude that national saving and domestic investment in the U.S. are not cointegrated.
This is also in contrast to most of the findings reported so far here.
4.3
Trivariate System
In the trivariate system with investment, saving, and output, the results indicate that these three
series are cointegrated with one cointegrating vector. This finding holds for all sample periods
covered in this study and for both test statistics (see Tables 4–7). Here we find cointegration using
the annual data also (see Table 3). This means that domestic investment and national saving are
indeed cointegrated, as the neoclassical growth theory predicts.
4.4
Cointegrating Vector Estimates
The estimated cointegrating vectors and the corresponding adjustment matrices for all
cointegration relationships found in this study are reported in Table 8. According to these estimates,
the long run coefficient on saving shows remarkable stability with the exception of the annual data
where the estimated coefficient is a little bit higher.23 The speed of adjustment figures reported in the
right hand side columns of Table 8, seems rather high, especially for the last decade. This suggests
that investment and saving adjust fast through time towards their long-run equilibrium levels.
In sum, Feldstein-Horioka puzzle may not be a puzzle at all because the results reported by
Miller (1988), Otto and Wirjanto (1989), and Gulley (1992) seem to be biased due to statistical
misspecification.24 More importantly, since my finding is robust across all sample periods
22 When we use the annual data, the null of no cointegration cannot be rejected (see Table 3).
23 Sinn (1992) also finds that the estimated saving coefficient is higher for lower frequency data.
12
considered, I conclude that it is unlikely that investment-saving correlations will provide any
information on the true degree of capital mobility.25 In the next section I address the question of
how the cointegration test result can be interpreted in the framework of a modern dynamic
optimization model of open economy.
5.
The Implications of Intertemporal Budget Constraint in Open Economy
Modern dynamic models of open economy also predict that investment and saving will be
correlated in the long run. To see this, consider a dynamic intertemporal optimization model of open
economy, where the central planner faces a budget constraint of the form
d Bt
= ρ t Bt + Ct + It + Gt – Yt
dt
(4)
where B denotes foreign debt, C denotes consumption, I denotes investment, G denotes
government expenditures, Y denotes output, and ρt denotes time varying world interest rate.
According to (4), the change in foreign debt equals spending minus production, where spending is the
sum of consumption, investment, and government spendings, and interest payments on the existing
debt. The central idea of this intertemporal budget constraint is that an open economy may borrow
from abroad to pay for the excess of spending over production, or it may lend to a foreign country to
accommodate the excess of production over spending. Thus, the existence of world capital market
enables the economy to accommodate temporary imbalances between production and spending.
The intertemporal budget constraint given in (4) is actually a nonhomogenous first order linear
differential equation with a time varying coefficient. To derive the long-run implication of this budget
constraint, integrate it forward by solving the differential equation for Bt. This yields
Bt =
AΨ t–1
+
Ψ t–1
∞
t
Ψ s Y s – Cs – Is – Gs ds
(5)
24 As already mentioned in section 3, the integration test results of Miller (1988) and Gulley (1992), and consequently
their cointegration tests, may not be valid because the tests these authors use are not appropriate.
25 I have also tested for cointegration between I and S measured in ratios under the assumption that the series contain
no deterministic trend. The results are the same. I shall also mention that the investment-saving relationships
estimated here were also estimated recursivly and the results (not reported) indicate a significant parameter constancy.
13
where A is an arbitrary constant which can be set to zero for simplicity without affecting the main
t
result, and Ψt ≡ e –
0
ρs ds
is the discount factor applied to the returns of time t period into the future.
s
In other words, a dollar at time t has a present value of Ψt . Similarly, Ψs ≡ e–
0
t+ s
ρv dv
=e –
t
ρv dv
,
which is used in deriving (5). The discount factor Ψt –1 Ψs gives the time t value of a dollar to be
1
delivered at time s. 26
Now assume that the t lim
Ψ t Bt = 0. This imposes a restriction that the representative agent
→∞
will not be able to borrow forever. Without this constraint the budget constraint given in (4) is not
really constraining since without it the optimal strategy for the agent to pursue would be to maximize
consumption by continuously borrowing as much as possible and accumulate ever increasing debt.
The above assumption prevents from the representative agent to choose such an explosive path of
borrowing. At the same time, however, the assumption does not impede the agent’s ability to incur a
temporary debt to accommodate temporary imbalance between production and spending.27
∞
Rearranging (5), and after adding and subtracting the expression
t
Ψs Ts ds to and from its
right hand side, where T denotes government revenues, we get
∞
t
Ψs Isds =
∞
t
Ψ s Ys – Ts – C s + Ts – Gs ds.
(6)
The first term in the brackets on the right hand side of (6) is the present discounted value of private
saving, while the second term is the present discounted value of public saving. Their sum equals
national saving, which I denote by S. Therefore,
∞
t
Ψs Isds =
∞
t
Ψ sSs ds.
(7)
The equality in (7) says that in the long run national saving and domestic investment will be correlated
regardless of the degree of capital mobility. Therefore, if investment and saving are cointegrated, it
will indicate that the country is solvent in the sense that it does not violate its intertemporal budget
26 To see that Ψ is indeed the discount factor, let us evaluate it for ρ = ρ, ∀s. This yields Ψ =e –
t
s
t
is the standard discount factor when interest rate is fixed.
27 This condition is sometimes called a no-Ponzi-game condition.
t
0
ρ ds
=e –ρ t , which
14
constraint. Thus, a test of cointegration of saving and investment is merely a test of country’s
economic solvency and it cannot be informative on the degree of international capital mobility.
Obstfeld (1991) reachs a similar conclusion based on a version of the model used by Trehan and
Walsh (1991).
Our findings based on cointegration test of investment and saving indicate that the series are
indeed cointegrated, that is, they are correlated in the long run, which suggests that the U.S. economy
is solvent since it does not violate its dynamic budget constraint. A similar finding is reported by
Trehan and Walsh (1991) who examine the question of the long-run sustainability of the U.S.
international indebtness by testing whether the U.S. net international investment position satisfies the
condition implied by the intertemporal budget constraint.
6.
Conclusion
Feldstein and Horioka’s (1980) finding that saving and investment tend to be correlated in the
long run has received significant attention in the literature. This is because Feldstein and his
coauthors express the view that the long-run investment-saving comovement is an indicator of an
international capital immobility. If this is true, then the findings reported in this paper would suggest
that capital was never mobile during the 1947–87 period.
However, as Baxter and Crucini (1993) note, most economists disagree with this
interpretation. It is difficult to defend this argument for several reasons. First, the restrictions
imposed on the international capital mobility has been declining over time in the world economy. This
is particularly true since early 70s, when many developed, and at lesser degree developing countries,
have abolished most capital restrictions. Second, the increased deregulation and integration of the
world financial markets’ is not compatible with the idea of declining capital mobility. For example, as
Purvis (1985) suggests, the extreme volatility of exchange rates since the abandonment of the Bretton
Woods system provides persuasive evidence of capital mobility—large pool of liquid assets are
switched in response to anticipations of exchange rate movements. And third, studies that measure
capital mobility directly using various PPP, interest rate differential, and other equilibrium conditions,
conclude that capital is very mobile and that capital mobility has been increasing over time. These
studies include Sachs (1981), Purvis (1985), Obstfeld (1986a, 1986b), Frankel and MacArthur
15
(1988), Popper (1990), Baxter and Crucini (1993), and Hutchison and Singh (1993). For example,
Hutchison and Singh (1993) examine real interest rate differential between the U.S. and Japan and
conclude that capital mobility is very high. Popper (1990) uses interest and currency arbitrage
conditions along with financial asset returns and finds that capital is as mobile in the long-run as is in
the short-run.
This paper argues that there is nothing mysterious in the investment-saving comovement.
Since the neoclassical growth theory predicts that in the steady state investment and saving will be
proportional to output and therefore will grow at the same rate, it would be surprising if we did not
find high long-run correlation between investment and saving. In the theoretical section of the paper it
is shown that the modern optimization-based dynamic model of open economy also predicts that
investment and saving will be correlated in the long run unless the economy violates its dynamic
budget constraint. This is because in an open economy the individuals face an intertemporal budget
constraint which allows for the possibility that they may incur debt during some periods of their lives
to smooth consumption. However, debt cannot be accumulated forever and therefore periods of
current account surpluses must be followed by periods of current account deficits, and visa versa.
This implies that in the long run investment and saving would be correlated, regardless of the degree
of capital mobility, as long as the intertemporal budget constraint is not violated. Therefore, if
investment and saving are cointegrated, then it indicates that the U.S. economy is solvent in the sense
that it does not violate its dynamic budget constraint. Thus, a test of cointegration of saving and
investment is merely a test of country’s economic solvency and it cannot be informative on the degree
of international capital mobility.
Recent studies by Miller (1988), Otto and Wirjanto (1989), and Gulley (1992) reexamine the
time series evidence on the bivariate investment-saving comovement using post-war U.S. data and
find that investment and saving are not cointegrated, which is contrary to the prediction of the
neoclassical growth model. This paper demonstrates that their finding may be caused by a simple
statistical misspecification, namely by omitted variables problem. Using annual and quarterly data
and improved cointegration methodology, I demonstrate that if output is added to the system and the
three variables are examined jointly, then they are always cointegrated, as predicted by theory. This
16
finding is true for fixed and flexible exchange rate periods.
As an additional evidence, note that if Feldstein-Horioka line of argument was valid, then the
huge capital inflow experienced by the U.S. during the first term of Reagan administration would
show up in a form of diminished long-run investment-saving correlation. The findings reported in
this paper, however, do not support this view. Therefore, I conclude that the observed long-run
correlation between investment and saving cannot be useful in measuring the degree of international
capital mobility.
17
6.40
8.30
6.20
8.10
6.00
7.90
5.80
7.70
5.60
7.50
5.40
7.30
5.20
7.10
5.00
1945
1950
1955
1960
I
1965
1970
Years
1975
1980
S
1985
6.90
1990
Y
Figure 1. Investment, Saving, and Output in Levels, U.S.,
Annual Data, 1947–87
0.20
0.19
0.18
0.17
0.16
0.15
0.14
0.13
0.12
0.11
1945
1950
1955
1960
I/Y
1965
1970
Years
1975
1980
1985
S/Y
Figure 2. Investment’s and Saving’s Share in the Output,
U.S., Annual Data, 1947–87
1990
18
6.50
8.30
6.25
8.10
7.90
6.00
7.70
5.75
7.50
5.50
7.30
5.25
7.10
5.00
6.90
6.70
4.75
1945
1950
1955
1960
I
1965
1970
Years
1975
1980
S
1985
1990
Y
Figure 3. Investment, Saving, and Output in Levels, U.S.,
Quarterly Data, 1947:1–87:3
0.22
0.20
0.18
0.16
0.14
0.12
0.10
1945
1950
1955
1960
I/Y
1965
1970
Years
1975
1980
1985
S/Y
Figure 4. Investment’s and Saving’s Share in the Output,
U.S., Quarterly Data, 1947:1–87:3
1990
19
Table 1. Time Series Properties of Investment, Saving, and Output: Annual Data†
Period
Series
1947–87
(n = 41)
I
S
Y
I/Y
S/Y
Miller’s
ADF ◊
Otto and
Wirjanto’s
ADF ◊
Gulley’s
ADF ◊
Corrected ADF statistic*
________________________
Levels
First differences
–1.47
–1.59
–1.22
–3.53 b
–3.38 c
–3.57 b
–3.71 b
–4.01 b
† Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. I used Box-
Pierce, Ljung-Box, and Lagrange Multiplier tests (not shown to save space) to verify that the error terms in all
the regressions reported in this paper are not serially correlated.
* The corrected ADF test statistics are estimated using equation (3). The 1%, 5%, and 10% significance
critical values for the corrected ADF statistics are –4.19, –3.52, and –3.19, respectively. These critical
values are tabulated in MacKinnon (1991).
◊ Miller (1988), Otto and Wirjanto (1989), and Gulley (1992) do not use the annual data.
20
Table 2. Time Series Properties of Investment, Saving, and Output: Quarterly Data†
Period
Series
1947:1–87:3º
(n = 163)
I
S
Y
I/Y
S/Y
1947:1–71:2§
(n = 98)
1971:3–87:3
(n = 65)
1980:1–87:3
(n = 31)
I
S
Y
I/Y
S/Y
I
S
Y
I/Y
S/Y
Miller’s
ADF ◊
Otto and
Wirjanto’s
ADF ◊
–3.98 b
–3.83 b
Corrected ADF statistic*
________________________
Levels
First differences
–0.86
–1.28
–3.33
–3.40
–2.03
–3.77 b
–3.75 b
–6.87 a
–6.90 a
–5.51 a
–3.22
–3.41
–1.35
–4.25 a
–4.11 a
–5.48 a
–5.19 a
–4.48 a
–2.66
–2.76
–2.43
–2.43
–2.58
–4.39 a
–4.49 a
–3.32 c
–4.43 a
–4.44 a
–2.99
–2.67
–1.75
–3.53 c
–3.16
–3.57 b
–3.63 b
–2.73
–3.70 b
–3.70 b
–4.88 b
–3.09 b
–0.14
–0.77
–3.54 b
–2.54
–0.46
–0.37
–3.91 b
–3.30 b
0.04
–0.40
–3.11
–3.25
–0.37
–0.74
Gulley’s
ADF ◊
–1.94
–2.78
–3.00 b
–1.22
I
S
Y
I/Y
S/Y
† Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. I used Box-
Pierce, Ljung-Box, and Lagrange Multiplier tests (not shown to save space) to verify that the error terms in all
the regressions reported in this paper are not serially correlated.
* The corrected ADF test statistics are estimated using equation (3). The 1%, 5%, and 10% significance
critical values for the corrected ADF statistics are –4.01, –3.43, and –3.14 for the period 1947:1–87:3,
–4.05, –3.45, and –3.15 for the period 1947:1–71:2, –4.10, –3.47, and –3.16 for the period 1971:3–87:3, and
–4.28, –3.56, and –3.21 for the period 1980:1–87:3, respectively. These critical values are tabulated in
MacKinnon (1991).
◊ These ADF test statistics and the corresponding critical values are reported in respective author’s papers.
º Here Otto and Wirjanto (1989) actually cover the period 1956:1–87:4.
§ Here Otto and Wirjanto (1989) actually cover the period 1956:1–71:2.
21
Table 3. Cointegration Test: Annual Data, 1947–87*
Critical Values
Sample
Size
Variables
Included
Test
H0
H1
n =41
I, S
λmax
r= 0
r≤ 1
r= 1
r= 2
6.56
1.69
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r= 2
8.26
1.69
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
λmax
r= 0
r≤ 1
r≤ 2
r= 1
r= 2
r= 3
20.18 c
6.23
2.97
20.96
14.06
3.76
18.59
12.07
2.68
21.07
14.90
8.17
18.90
12.91
6.50
JT
r= 0
r≤ 1
r≤ 2
r≥ 1
r≥ 2
r= 3
29.39 c
9.20
2.97
29.68
15.41
3.76
26.78
13.32
2.68
31.52
17.95
8.17
28.70
15.66
6.50
I, S, Y
Test
Statistic
Trend in DGP
95%
90%
No Trend in DGP
95%
90%
* Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The critical values are
taken from Osterwald-Lenum (1992).
Table 4. Cointegration Test: Quarterly Data, 1947:1–87:3*
Critical Values
Sample
Size
Variables
Included
Test
H0
H1
Test
Statistic
Trend in DGP
95%
90%
n =163
I, S
λmax
r= 0
r≤ 1
r= 1
r= 2
18.58 b
2.39
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r= 2
20.97 b
2.39
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
λmax
r= 0
r≤ 1
r≤ 2
r= 1
r= 2
r= 3
27.05 b
8.66
5.79
20.96
14.06
3.76
18.59
12.07
2.68
21.07
14.90
8.17
18.90
12.91
6.50
JT
r= 0
r≤ 1
r≤ 2
r≥ 1
r≥ 2
r= 3
41.51 b
14.46
5.79
29.68
15.41
3.76
26.78
13.32
2.68
31.52
17.95
8.17
28.70
15.66
6.50
I, S, Y
* See the footnote underneath Table 3.
No Trend in DGP
95%
90%
22
Table 5. Cointegration Test: Quarterly Data, 1947:1–71:2
Critical Values*
Sample
Size
Variables
Included
Test
H0
H1
n =98
I, S
λmax
r= 0
r≤ 1
r= 1
r= 2
9.45
0.82
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r= 2
10.28
0.82
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
λmax
r= 0
r≤ 1
r≤ 2
r= 1
r= 2
r= 3
32.08 b
10.49
0.02
20.96
14.06
3.76
18.59
12.07
2.68
21.07
14.90
8.17
18.90
12.91
6.50
JT
r= 0
r≤ 1
r≤ 2
r≥ 1
r≥ 2
r= 3
42.60 b
10.52
0.02
29.68
15.41
3.76
26.78
13.32
2.68
31.52
17.95
8.17
28.70
15.66
6.50
I, S, Y
Test
Statistic
Trend in DGP
95%
90%
No Trend in DGP
95%
90%
* See the footnote underneath Table 3.
Table 6. Cointegration Test: Quarterly Data, 1971:3–87:3*
Critical Values
Sample
Size
Variables
Included
Test
H0
H1
Test
Statistic
Trend in DGP
95%
90%
n =65
I, S
λmax
r= 0
r≤ 1
r= 1
r= 2
19.44 b
4.91
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r= 2
24.35 b
4.91
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
λmax
r= 0
r≤ 1
r≤ 2
r= 1
r= 2
r= 3
22.39 b
7.24
4.21
20.96
14.06
3.76
18.59
12.07
2.68
21.07
14.90
8.17
18.90
12.91
6.50
JT
r= 0
r≤ 1
r≤ 2
r≥ 1
r≥ 2
r= 3
33.85 b
11.45
4.21
29.68
15.41
3.76
26.78
13.32
2.68
31.52
17.95
8.17
28.70
15.66
6.50
λmax
r= 0
r≤ 1
r= 1
r= 2
15.09 b
1.84
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r≥ 2
16.93 b
1.84
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
I, S, Y
I, S
Y Y
* See the footnote underneath Table 3.
No Trend in DGP
95%
90%
23
Table 7. Cointegration Test: Quarterly Data, 1980:1–87:3
Critical Values*
Sample
Size
Variables
Included
Test
H0
H1
Test
Statistic
Trend in DGP
95%
90%
(n = 31)
I, S
λmax
r= 0
r≤ 1
r= 1
r= 2
17.48 b
3.08
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r= 2
20.57 b
3.08
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
λmax
r= 0
r≤ 1
r≤ 2
r= 1
r= 2
r= 3
27.23 b
11.84
1.80
20.96
14.06
3.76
18.59
12.07
2.68
21.07
14.90
8.17
18.90
12.91
6.50
JT
r= 0
r≤ 1
r≤ 2
r≥ 1
r≥ 2
r= 3
40.88 b
13.65
1.80
29.68
15.41
3.76
26.78
13.32
2.68
31.52
17.95
8.17
28.70
15.66
6.50
λmax
r= 0
r≤ 1
r= 1
r= 2
20.34 b
1.29
14.06
3.76
12.07
2.68
14.90
8.17
12.91
6.50
JT
r= 0
r≤ 1
r≥ 1
r≥ 2
21.64 b
1.29
15.41
3.76
13.32
2.68
17.95
8.17
15.66
6.50
I, S, Y
I, S
Y Y
* See the footnote underneath Table 3.
No Trend in DGP
95%
90%
24
Table 8. The Estimated Normalized Cointegrating Vectors and Corresponding Adjustment Matrices*
Cointegrating Vector β
Sample
Period
I
S
Y
Annual
1947–87
–1.00
1.19
–0.09
Quarterly
1947:1–87:3
–1.00
1.00
Quarterly
1947:1–87:3
–1.00
1.14
Quarterly
1947:1–71:2
–1.00
1.06
Quarterly
1971:3–87:3
–1.00
1.11
Quarterly
1971:3–87:3
–1.00
1.14
–1.00
1.10
Quarterly
1980:1–87:3
–1.00
1.05
I
S
Y
–0.33 –0.63
–0.11
–0.46
–0.03
–0.27
–0.48 –0.45
0.14
0.46
0.42
–0.80
–1.08
–0.41
–0.70 –0.89
–0.01
–1.00
Quarterly
1980:1–87:3
S
Y
–0.18
Quarterly
1971:3–87:3
Quarterly
1980:1–87:3
I
Y
Adjustment Matrices α
–1.00
1.05
S
Y
–0.36
–0.77
–2.47
–3.40
0.21
1.06
–0.01
I
Y
–2.87
–3.22
–2.25
–3.39 –2.88
* The normalization is done by setting the coefficient on investment (I) to –1.00. The cointegrating vectors and
the adjustment matrices presented here correspond to the cointegration relationships established in Tables 3–7
and are presented in the same order as the tables.
25
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