Daniel Levy

Shomu Banerjee

Bob Carpenter

Paul Romer

Martin J Bailey

Mark Bergen

Hashem Dezhbakhsh

Hal Fried

Maxwell Fry

Amihai Glazer

Jinook Jeong

Roger Kormendi

David Lilien

Giovanna Mossetti

Will Roberds

Shelton Schmidt

Output, Capital, and Labor in the Short, and Long-Run*

Keywords: O47, E22, E24, E32 Growth Accounting, Capital Investment, Output Fluctuation, Employment, Business Cycles and Aggregate Fluctuation, Frequency Domain Analysis, Spectrum and Cross-Spectrum, Coherence, Phase Shift, Gain, Zero-Frequency, Capital and Labor Elasticity of Output, Short-Run, Long-Run, Capital'

Using a new series of capital stock and frequency domain analysis, this paper provides new empirical evidence on the relative importance of capital and labor in the determination of output in the short and long-run. Contrary to the common practice in the traditional growth accounting literature of assigning weights of 0.3 and 0.7 to capital and labor inputs respectively, the evidence presented here suggests that capital is a far more important factor than labor for determination of output at and near the zero frequency band. Furthermore, I show that the zero-frequency labor elasticity of output may well be close to zero, or even zero. Additional findings reported here support the traditional accelerator model of investment as a good description of the long-run investment process.

I. Introduction

In traditional growth accounting calculations that originated from Solow's [32; 33] seminal work, macroeconomists usually assign output elasticities of 0.30 and 0.70 to capital and labor inputs respectively. These values are based on the assumption that producers are operating in a competitive, profit maximizing, constant returns to scale environment in which factors of production are paid their marginal product. Brown [START_REF] Brown | On the Theory and Measurement of Technological Change[END_REF], Douglas [START_REF] Douglas | Are There Laws of Production?[END_REF], and

Intriligator [START_REF] Intriligator | Embodied Technical Change and Productivity in the U.S, 1929-1958[END_REF] have provided empirical support for these assumptions for the pre-war and inter-war periods. Their estimated capital and labor elasticities of the output were around 0.25 and 0.75 respectively.

However, a recent study by Paul Romer [START_REF] Romer | Crazy Explanations for the Productivity Slowdown[END_REF] concludes that the contribution of capital accumulation to long-run growth is substantially underestimated in the conventional growth accounting analysis and that the true capital elasticity of output may actually be greater than its share in total income because of positive externalities associated with investment. On the other hand, Romer suggests that the contribution of labor is considerable overestimated and that the true labor elasticity may actually be smaller than its share in income because of negative externalities associated with labor. In particular, according to Romer's estimation, the long-run capital and labor elasticities of output probably lie in the range of 0.7-1.0 and 0.1-0.3 respectively. But his estimates come from historical data of output, capital, and labor averaged over 10-and 20-year intervals for the periods 1890-1980 and 1839-1979, respectively. Therefore, as Romer himself suggests, the signal-to-noise ratio may be too small for a sensible interpretation of these figures, since the above time series contain only 7-9 observations. In addition, a long-run averaging of the data may not have completely eliminated the effect of business cycle fluctuations.

Bernanke [START_REF] Bernanke | Comments on Romer (1987)[END_REF]204] expresses doubts about the correctness of Romer's estimates since "it cannot literally be true that output is independent of labor input, [and therefore] this result must be caused by an estimation bias."

The importance of capital accumulation in the growth of the U.S. economy is emphasized in other studies that analyze the sources of long-term growth in the U.S. economy [4; 5; 8; 10; 22; 23]. For example, Jorgenson [START_REF]Productivity and Postwar U.S. Economic Growth[END_REF][START_REF] Levy | Aggregate Output, Capital, and Labor in the Post-War U.S. Economy[END_REF] argues that "comparing the contribution of capital input with other sources of output growth for the period 1948-1979 as a whole makes clear that capital input is the most significant source of growth." Denison [START_REF] Denison | The Contribution of Capital to Economic Growth[END_REF]220] makes a similar argument: "I do not share the other extreme view, sometimes encountered, that capital can be ignored because its significance is hard to establish if one fits a production function by correlation analysis. I stress again: capital is an important growth source. It has sometimes contributed importantly to differences in growth rates between periods and places. More capital formation would raise the growth rate." 1 From a theoretical point of view, this argument is not really new. The usual assumption used in standard microeconomic models that the stock of capital is fixed in the short-run but variable in the long-run implies that variations in the stock of capital will affect the output in the long-run. 2 More importantly, dynamic models that involve some kind of transaction costs usually make similar predictions. 3 The purpose of this paper is to provide new empirical evidence on the relative importance of capital and labor in the determination of output in the short-and long-run.

Unlike the studies cited above, the methodology applied here uses frequency domain analysis. The advantage of using the frequency domain framework is that it allows us to conduct the analysis on a frequency-by-frequency basis for describing empirical cyclical regularities in the data and examining the dynamic relationship between time series without the intervention of an econometric model. The frequency domain methodology used here is nonparametric and therefore requires no behavioral or distributional assumption about the time series of output, capital, and labor. The only requirement is that the series analysed be stationary. The quarterly time series of capital stock used here was constructed recently and thus differs from the data used by Romer [START_REF] Romer | Crazy Explanations for the Productivity Slowdown[END_REF] and others. Despite these differences, the findings reported in this paper indicate that capital indeed is a far more important factor than labor in the determination of output at the zero frequency band. Furthermore, I show that the zero-frequency labor elasticity of output may well be close to zero, or even zero. An additional finding of this paper is related to the accelerator model of investment: it turns out that output leads capital at the zero frequency band which suggests that the traditional accelerator model may be a good description of the long-run investment process.

Statistical evidence supporting these ideas are derived below by examining the capitaloutput and the labor-output relationships using cross-spectral analysis. Spectral analysis provides a useful framework for studying the issues raised here because in the frequency domain short and long-run relationships between time series can be characterized and analysed by looking at the behavior of the series in the high and low-frequencies, respectively. The main disadvantage of ordinary time domain regression analysis in the context discussed here is the fact that it implicitly treats all frequencies equally. In addition, as Chow [START_REF] Chow | Econometrics[END_REF], Harvey [START_REF] Harvey | Time Series Models[END_REF], and many others argue, although the information gained from frequency domain analysis is theoretically a transformation of its time domain analog, some dynamic and cyclical features of the data are easier to identify and interpret in the frequency domain.

The paper is organized as follows. In the next section, I briefly review the statistical methodology used in this study. In section III, I describe the data set. Next, in section IV, I present and discuss the empirical results of the study. In section V, I discuss the implications of the findings for growth accounting in the context of U.S. business cycles. The paper ends with a brief summary of the main results and some concluding remarks.

II. The Methodology

Spectral analysis makes it possible to conduct time series analysis in the frequency domain, where we think of a stationary series as being made up of sine and cosine waves of different frequencies and amplitudes. In a univariate case, we are interested in determining how much of the total variance ('power') of the series is determined by each frequency component. In a bivariate setup, spectral analysis provides a description of a linear relationship between time series at different frequencies. 4 For a covariance stationary univariate process y t , the autocovariance function is given by the expression

γ (s) = E y t + s -µ y t -µ ( 1
)
where µ is the mean of the process. It is usually assumed that both γ (s) and µ are time independent which is essential for past observations to be useful in describing the present or the future. It follows that γ (s) = γ (-s). The spectrum of the series y t is defined as the Fourier transform of its autocovariance function, and is given by

f y (ω) = (1/2π) γ (s) e -isω ∑ s =-∞ ∞ 0 ≤ω ≤ π ( 2
)
where ω is the frequency and is measured in cycles per period (in radians).

For a bivariate covariance stationary process y t , x t , the cross covariance function given by

γ yx (s) = E y t + s -µ y x t -µ x , (3)
measures the degree of linear association between the two stochastic processes for different time lags and is independent of time. The cross spectrum is the Fourier transform of the cross covariance function and is given by

f yx (ω) = (1/2π) γ yx (s) e -isω ∑ s =-∞ ∞ (4)
which is a complex-valued function of ω. Since the cross spectrum as given above cannot be examined directly, the usual practice is to compute and plot 'squared coherence', 'phase', and 'gain'. The squared coherence which is given by

C yx (ω) = f yx (ω) 2 / f y (ω) f x (ω) 0 ≤ C yx (ω) ≤ 1 (5)
is analogous to the square of the correlation coefficient between the series at each frequency.

That is, it represents the degree to which one time series can be represented as a linear function of the other. The higher the C yx (ω), the more closely related are the two series at frequency ω. By its construction the coherence says nothing about the sign of the relation between the two series, nor anything about the timing of any lead/lag in the relation. 5 The phase, P yx (ω), is a measure of the phase difference or the timing between the frequency components of the two series. It is measured in the fraction of a cycle that x leads y and is given by

P yx (ω) = (1/2π) arctan -Im f yx (ω) / Re f yx (ω) , (6)
where Im and Re are the imaginary and real parts of the cross spectrum. It is worth noting that determining lead-lag relationship using the phase differs from the method used by the National Bureau of Economic Research (henceforth NBER). NBER determines the lead-lag relationship using only peaks and troughs without considering non-turning point periods.

The phase, however, takes into account all the time points for which the data are available.

The phase is known only up to adding or substracting an integer since adding or substracting one whole cycle to an angle will not change its tangent. 6 The gain indicates how much the spectrum of y t has been amplified or attenuated to approximate the corresponding frequency component of x t . It is essentially the regression coefficient of the process y t on the process x t at frequency ω and is given by

G yx (ω) = f yx (ω) / f y (ω) ≥ 0. (7)
A small gain at frequency ω indicates that x has little effect on y at that frequency. 7 In sum, we may interpret coherence, phase, and gain in terms of the ordinary regression analysis terminology if we imagine running a regression equation of y on x at each frequency. The squared coherence is the frequency domain analogue of the time domain correlation coefficient, R 2 , and is calculated at each frequency. The regression coefficient is just the gain if there is no time lag between y and x. If there is a time lag, the gain can be interpreted as the regression coefficient if the series were lagged just the right amount to eliminate any phase shift, and the phase is the angle by which they would have to be shifted.

An additional set of statistics crucial for a reliable interpretation of spectral analysis results include the significance test statistics associated with coherence, phase and gain.

Unfortunately, they are rarely reported in published articles. I have computed these statistics and they are reported along with the estimates of coherence, phase, and gain. As a significance test for squared coherence, I test whether H 0 : C yx (ω) = 0. For phase and gain, I provide a 95% confidence interval. 8

III. The Data

The quarterly time series used in this study consists of real output, real capital stock, and employment data for the period 1948-1983 (144 observations). 9 While there is nothing unique about the output, and employment data, the quarterly stock of capital data was recently constructed by Balke and Gordon [START_REF] Balke | Historical Data[END_REF] by using the corresponding annual capital stock data published in the Survey of Current Business. This was done by treating the annual series' values as the beginning and the ending values of the quarterly series using the fact that under a fixed exponential rate of depreciation, the quarterly series satisfy

K t = I t + 1 -δ K t -1 .
The depreciation rates (δ ) of each capital stock component were iterated until the fourth quarter's value converged to the end of the year value from the annual series. The resulting estimated annual depreciation rates for the stock of nonresidential structures and producers' durable equipment are 6.036% and 14.3%, respectively. 10 The quarterly employment figures are taken from Business Conditions Digest, Bureau of Economic Analysis, February 1984, p. 101.

IV. Results of Spectral Analysis

Since spectral analysis methodology outlined here applies only to stationary processes, most economic times series require some kind of filtering prior to spectral analysis as they usually tend to be nonstationary. In order to determine whether the data I use is stationary or not, I formally examine the unit root properties of the time series of output, capital, and labor.

Recall that, if a time series x t has to be differenced d times to make it stationary, then we say that x t is integrated of order d, x t ~I(d). The number of differencing needed to make a time series stationary corresponds to the number of unit roots the series contains. Therefore in order to determine whether the time series of output, capital, and labor are stationary, I examine each series for presence of unit roots using the Augmented Dickey-Fuller (henceforth ADF) univariate tests of the form

∆ x t = α 0 + α 1 t + γ x t -1 + φ i ∆ x t -i Σ i =1 4 + ε t , (8)
where x t is the series we are examining and t is a linear time trend. Engle and Yoo [START_REF] Yoo | Cointegrated Economic Time Series: A Survey with New Results[END_REF] and MacKinnon [START_REF] James | Critical Values for Cointegration Tests[END_REF] recommend including a linear trend component in the test equation to avoid the dependence of the test statistic's distribution on the true value of α 0 . The null hypothesis is that γ = 0, which means that the series contains an unit root and is thus nonstationary, i.e.,

x t ~I(1). The alternative hypothesis is that the series are stationary, that is, x t ~I(0).

Initially, I test the hypothesis of an unit root in the series measured in levels. The results are reported in the first column of Table I. As the values of the ADF t-statistics indicate, the hypothesis of nonstationarity cannot be rejected for either of the series. Next, I test whether first differences of the same series are nonstationary. Based on the figures presented in the last column of Table I, the hypothesis of an unit root in the differenced series can be rejected with 1% significance. Therefore, I conclude that all three series are I(1), and thus can be represented as difference stationary processes. Consequently, the time series of output, capital, and labor are all log-differenced prior to the application of spectral analysis. 11 Next, using the methodology described in Section 2, I estimate the squared coherence, phase, and gain between the time series of real GNP and real stock of capital on one hand, and between the real GNP and employment level on the other. 12 The results are reported in Figures 123456. On these figures the frequency along the horizontal axis, ω, is measured in radians. i.e., 0 ≤ ω ≤ π. Each frequency corresponds to a particular periodicity (or a cycle length) according to the mapping, ξ = 2π/ω, where ξ denotes the length of a cycle. 13 In the analysis that follows, the frequency range 0 ≤ ω ≤ π is divided into long-run, business cycle, and short-run frequency bands. The cut-off points of these frequency bands are identical to those used in the modern business cycle literature. For example, according to Englund et al.

[15] and Hassler et al. [START_REF] Hassler | The Swedish Business Cycle: Stylized Facts Over 130 Years[END_REF], most students of business cycles and growth define business cycles as 12-32 quarter cycles. Their estimate of the average length of a business cycle is about 20 quarters, which corresponds to the frequency of ω =0.31. The frequencies below business cycle frequency band correspond to the long-run, while the frequencies above business cycle frequency band correspond to the short-run. The shortest identifiable cycle is a two-period cycle and it corresponds to frequency ω =π , also known as the Nyquist frequency.

The estimated squared coherence between the real GNP and the real stock of capital is plotted in Figure 1. The horizontal line at 0.31 is the 95% critical value derived from testing the hypothesis, H 0 : C yx (ω) = 0. Thus, any coherence value above 0.31 is statistically bigger than zero. 14 As the figure shows, the squared coherence between real output and the stock of capital is statistically significant at most frequency bands including business cycle frequencies, 0.19 ≤ ω ≤ 0.51, which correspond to cycles of 12-32 quarters length. In particular, the coherence is statistically significant at the zero frequency band, suggesting a long-run relationship between capital and output. The coherence is not statistically significant at the frequency bands 1.98-2.23, and 3.06-3.14, which corresponds to a 2-3 quarter cycle. The coherence is highest at the frequency corresponding to 12 quarter cycle.

The phase of the real GNP with the stock of capital is provided in Figure 2 along with the 95% confidence interval. Note that the confidence interval is smaller, the higher the squared coherence. 15 The phase diagram indicates an upward trend over the frequency band 0 ≤ ω ≤ 0.29, which corresponds to cycles of about five years and longer. Then it remains relatively stable up to about ω =0.76, followed by another downward trend up to about ω =1.57, and then an upward trend up to ω =1.79. In the range of frequencies 2.25 ≤ ω ≤ π, the phase indicates another downward trend. 16 A visually observable trend in the phase diagram usually indicates a fixed time lead-lag structure which implies that there is a fixed phase differential between the series at the corresponding frequency band. When there is a time delay, the phase is a linear function of frequency, the slope representing the magnitude of the delay. Thus, the fixed time delay can be estimated by approximating the slope of the phase trend at the frequency band. For instance, a perfect one-period lag relationship would result in a straight trend line with a slope of one (radian per radian). In case of a horizontal phase at some constant value (e.g., at the frequencies 0.29 ≤ ω ≤ 0.76), the lead-lag relationship is variable (fixed angle lag). That is, the smaller the frequency, the larger the time lag between the corresponding components.

Following these guidelines, the estimated phase diagram of Figure 2 suggests the following: at the zero-frequency band (specifically for ω ≤ 0.29), the real GNP seems to lead the capital stock by about 6 quarters (1.71/0.29). Afterwards, the lead-lag mechanism seems to be varying with frequency. Along business cycle frequencies (12-32 quarters), the smaller the frequency, the larger is the time-lead of the real GNP over the capital stock. In the range of frequencies 2.25 ≤ ω ≤ π, the phase indicates a downward trend, which implies that capital stock leads real GNP. The diagram indicates several phase shifts (including at the seasonal frequency, 4 quarters). Overall, the lead-lag relationship between real GNP and capital stock is complicated and varies with frequencies. The finding that output leads capital in the long-run, that is, at the zero frequency band, suggests that the traditional accelerator model of investment may be a good description of the long-run investment process.

The gain of real GNP with capital stock is plotted in Figure 3 along with 95% confidence interval. As the diagram indicates, the gain starts at about 0.36 at the zero frequency band and keeps declining continuously all the way to about 0.03-0.04 at the shortrun frequency band. This means that the increase in output associated with an increase in the stock of capital is much larger in the long-run than in the short run.

In sum, cross spectral analysis of real output and capital stock suggests that there is a significant correlation between output and capital across almost the entire frequency band.

However, in magnitude, this relationship is much more important in the long-run than in the short-run. The implied lead-lag relationship supports the accelerator model of investment as a good description of the U.S. long-run investment process. for ω ≤ 0.12, which corresponds to 13 year or longer cycles), there is no statistically significant correlation between output and employment. The correlation is not significant also at 2.08 ≤ ω ≤ 2.35, which corresponds to about a three-quarter cycle. But there is a sharp increase in the coherence immediately after ω =0.12, with the peak occurring at ω =0.39, which corresponds exactly to a 16 quarter cycle. This obviously identifies the output-labor comovement at this frequency as a typical business cycle phenomenon.

The phase diagram plotted in Figure 5 shows that the phase is relatively stable at the frequencies, ω ≤ 2.09, fluctuating around zero. This is an indicator of a stable contemporaneous relationship between output and labor at frequencies that correspond to 3 quarter and longer cycles. That is, output and labor are in phase in the long-run as well as across business cycles (12-32 quarters). The diagram also has a negative trend for ω ≤ 2.52, which suggests that in the very short-run, employment leads output by about 6 quarters [2.52-(-1)]/(3.14-2.52). 17 The gain plot of output with employment suggests a relatively stable relationship across almost the entire frequency band. At frequencies close to zero (specifically, for ω ≤ 0.12, which corresponds to 13 year or longer cycles), its value is not statistically significant, which again suggests that in the zero-frequency band, labor does not matter for the determination of output. That is, the zero-frequency labor elasticity of output seems to be zero. At the frequencies where the coherence is not statistically significant, the confidence interval of the gain contains zero, because the lower the coherence, the larger the sample variance of the estimated gain. Although the gain attains a maximum at ω =2.20, which corresponds to a three quarter cycle, its confidence interval is very wide at that frequency as the squared coherence is low, and thus the gain's estimate at that frequency is very imprecise.

In sum, the cross-spectral statistics indicate that the zero-frequency labor elasticity of the output is very small and may even be zero. Across other frequency bands (except at very short-run frequencies) output and labor are in phase, moving contemporaneously. 18 At the short-run frequencies, labor leads output by about 6 quarters. Overall, the relationship between labor and output is far more stable than between capital stock and output.

V. Implications for Growth Accounting

As shown in the previous section, at business cycle frequencies, that is at frequencies that correspond to 3-8 year cycles, the coherences between output and labor input and between output and capital stock are both high, suggesting that labor as well as capital are procyclical.

This finding is in line with the general findings documented by Burns and Mitchell [START_REF] Burns | Measuring Business Cycles[END_REF] about durations of various business cycles in the U.S. As Table II indicates, the American business cycles from 1854 to the present have varied in length between 2-10 years. The average length of cycle has been about 4-5 years, with the most common length being 3-4 years.

In Table III which is slightly higher than the average length of a business cycle. On the other hand, at and immediately near the zero frequency band (say, for ω <0.20) the ratio is very low, which indicates that at these frequencies the contribution of capital to output far exceeds the contribution of labor. This puzzling result obviously leads to the following question: which frequencies should be used for the determination of the relative weights of capital and labor inputs in growth accounting equations?

From the theoretical point of view, the classical model considers a state of the economy at a point in time, but under the assumption that prices have adjusted to clear markets.

Solow's [START_REF] Solow | A Contribution to the Theory of Economic Growth[END_REF] growth model, on the other hand, explains the growth pattern of an economy over many decades. Capital stock is a key factor in that model. But, capital stock is a slow moving variable as changes in it require the building of new factories and production lines, new machines, new structures, etc. Therefore, in the context of Solow's model, which is what I believe most students of growth and business cycles have in mind when they think of long-run growth, a period of just a few years might be considered short-run, since it may take many, many years for an economy to adjust to its steady-state equilibrium.

This obviously leads to the well-known discussions about the difficulties practitioners face with defining and measuring long-run trends. According to Kydland and Prescott [START_REF] Kydland | Business cycles: Real facts and Monetary Myth[END_REF][START_REF] Christensen | Measuring the Performance of the Private Sector in the U.S. Economy[END_REF] "the trend component for real GNP should be approximately the curve that students of business cycles and growth would draw through a time plot of this time series." This definition of long-run trend identifies growth time trend as a very low frequency phenomenon. The long-run then might be measured in decades rather than years. Therefore, in order to determine proper weights of capital and labor in growth accounting equations, we need to look at zero and near-zero frequencies. Consequently, based on the findings reported here, it follows that the conventional weights of capital and labor used in traditional growth accounting equations indeed overestimate labor's share and underestimate capital's share in national income.

VI. Summary and Conclusions

The purpose of this paper was to compare and contrast relative importance of labor and capital inputs for the determination of output in the short, and long-run. The findings indicate that at the frequency band, ω ≤ 0.12, which corresponds to cycles of 13 years or longer, capital is statistically important for the determination of output, while the level of employment does not seem to matter. That is, the zero-, and near zero-frequency elasticity of output with respect to labor seem to be close to zero. In the short run, the opposite is true:

the variations in output due to a given change in labor is much bigger than due to a change in capital. Thus, the statistical evidence provided here supports the view that in conventional growth accounting analysis the contribution of capital accumulation to long-run growth is indeed underestimated, while the contribution of labor is substantially overestimated.

The analysis of the lead-lag structure of capital-output and labor-output relationships suggest that across most frequency bands, output and labor are in phase, moving together contemporaneously. At the short-run frequencies, labor leads output by about 6 quarters.

The lead-lag structure of capital and output varies with the frequency band. In the long-run, output leads the stock of capital as predicted by the accelerator model of investment.

Thus, this paper contributes to the literature on the sources of growth of the U.S.

economy by showing that stock of capital plays a far more important role in the determination of output in the long-run than in the short-run, while employment level has a significant effect on output in the short, and medium-run, but not in the long-run.

3. It is interesting to note that historically Marshall [START_REF] Marshall | The Principles of Economics[END_REF] was the first to formally distinguish between different time periods in the context of the intertemporal production process. He actually considered four different time periods: (i) the period where all inputs are fixed ('very short-run'); (ii) the period where inputs are variable, and thus supply can be increased up to the highest level possible for given capital stock ('short-run'); (iii) the period where all the inputs are variable given the available technology ('long-run'); and (iv) the period where even the technology is variable ('very long-run').

4. Historically, spectral analysis seems to have originated in the work of Schuster [START_REF] Schuster | On Hidden Periodicities[END_REF], who has developed the method of periodogram analysis for finding hidden periodicities in sunspot data. The development of the modern formal theory of spectral analysis was initiated by Wiener [START_REF] Wiener | Generalized Harmonic Analysis[END_REF]. It was initially applied to engineering and physical science data where large data sets are generated by experiments, and was imported to economic time series data much later. The description of spectral methodology presented here follows Engle [START_REF] Engle | Interpreting Spectral Analysis in Terms of Time-Domain Models[END_REF], Fishman [START_REF] Fishman | Spectral Methods in Econometrics[END_REF], and Jenkins and Watts [START_REF] Jenkins | Spectral Analysis and its Applications[END_REF].

5. A useful property of coherence is its invariance under linear filtering. This means that the degree of linear association between time series as measured by coherence is preserved under linear filtering of the time series.

6. A similar ambiguity is present in time domain cross-correlation analysis if both negative as well as positive cross-correlation coefficients are found.

7. Note that the squared coherence may be constant over the entire frequency range.

However, as Fishman [START_REF] Fishman | Spectral Methods in Econometrics[END_REF] notes, the corresponding gain function will not necessarily be so because even if the same linear association exists at all frequencies, some frequency components may still be amplified or attenuated more than others.

8. The statistics used in these tests are provided by Jenkins and Watts [START_REF] Jenkins | Spectral Analysis and its Applications[END_REF].

9. The sample period stops in 1983 since the capital stock data series used here is not available thereafter.

10. See Balke and Gordon [START_REF] Balke | Historical Data[END_REF] for more details. Note that these figures are much higher than the depreciation rates used by the Bureau of Economic Analysis (BEA) in constructing its National Income and Product Accounts' (NIPA) estimates. According to John Musgrave of the BEA (personal communication), the BEA's estimated annual depreciation rates of the stock of nonresidential structures and producers' durable equipment are 1.5% and 6.0%, respectively. It is not clear to me why there are such large differences between the two estimates. It may be that because of the iterative nature of the estimation method Balke and Gordon [START_REF] Balke | Historical Data[END_REF] use, their estimate is nonlinear, while the BEA's estimates are constructed using a linear life time depreciation path.

11. The difference filter has been commonly used in frequency domain literature even before the recent development of unit root literature since it turns out that spectral representations of the original and the differenced series are related. An additional advantage of difference filter is the fact that it belongs to an important class of symmetric digital filters called nonnegative definite filters. These filters have zero phase shift for all frequencies and therefore passing time series through them does not alter the lead-lag relationship between the time series.

12. Various statistics (coherence, phase, and gain) reported in this study were all computed by first constructing the cross periodograms, which then were smoothed in order to get consistent estimates of the series' cross spectral densities. The smoothing was done by averaging the neighboring periodogram ordinates using a flat window with a width of 9. That is,

f yx (ω k ) = 1/9 I yx (ω k -i ) Σ i =-4 4
, where I yx is the estimated cross-periodogram, and k = 128 is the 20 number of ordinates.

13. Thus the frequency ω =0.524, e.g., corresponds to a three-year cycle if quarterly data is used.

14. The 99% critical value of coherence equals 0.43.

15. This is because the higher the coherence, the lower is the variance of the estimated phase.

16. I ignore frequencies at which the squared coherence is not statistically significant, as at these frequencies the gain is not statistically different from zero. Some portions of the confidence interval are not shown on Figures 2 and4 since at those frequency bands the phase's confidence interval is not defined. To see this, note that phase's confidence interval is given by

P yx (ω) ± arcsin [2/(τ -2)] F 2,τ -2 1-α [1 -C yx (ω)]/C yx (ω) 1/2
, where C yx (ω) and P yx (ω) are the estimated squared coherence and phase, respectively. Under the null hypothesis H 0 : C yx (ω) = 0, the quantity τ -2 C yx (ω) / 2 1 -C yx (ω) follows a Snedecor's F 2,τ -2 distribution.

Therefore, the expression under the square root will be greater than 1 for any

C yx (ω) ≤ C yx * ,
where C yx * is the 95% critical value of the coherence. However, arcsin is not defined if its argument is bigger than 1. Therefore, at frequency bands where the squared coherence is not statistically significant, the confidence interval of phase will be undefined. 17. I again ignore the frequencies at which the squared coherence is not statistically significant.

18. Baxter and King [START_REF] Baxter | Productive Externalities and Cyclical Volatility[END_REF][START_REF] Lau | Post-War Economic Growth in the Group-of-Five Countries: A New Analysis[END_REF] find that "major upward and downward movements in output and labor input occur together."

Figures 4 -

 4 Figures 4-6 provide the plottings of coherence, phase, and gain of the real output with employment. The coherence diagram reveals that at frequencies close to zero (specifically,

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. Coherence of Output and Labor, U.S., 1948-83

  , I have computed the ratio of the estimated output-labor to output-capital gain functions for ω ≤ 0.51. Comparison of the estimated gain functions at the business cycle frequencies indicates that the ratio of output-labor to output-capital gains is in the range

1.7-2.6. The commonly used ratio of labor and capital shares in income, 0.7/0.3 = 2.3, falls in this interval. As the figures presented in Table III indicate, the ratio of the estimated gains exactly equals 2.3 at the frequency ω =0.27, which corresponds to about a 6 year cycle,

Table I .

 I ADF Unit Root Test t-Statistics a _____________________________________________________________________ . The * indicates a significance at 1%. Critical values as tabulated in MacKinnon[START_REF] James | Critical Values for Cointegration Tests[END_REF], for α = 1, 5, and 10% are: -4.02, -3.44, and -3.14, respectively. The null hypothesis is H 0 : x~ I(1).

	Variable	ADF t-statistic	Variable	ADF t-statistic
	_____________________________________________________________________
	Output	-2.39	∆(Output)	-5.32 *
	Capital	-0.30	∆(Capital)	-4.05 *
	Labor	-1.62	∆(Labor)	-5.18 *
	_____________________________________________________________________

a

Table II .

 II Length of American Business Cycles, 1854-83 _________________________________________________________________________

	Length in Years	Number of Cycles
	_________________________________________________________________________
	2	2
	3	10
	4	9
	5	4
	6	2
	7	1
	8	1
	9	0
	10	1
	_________________________________________________________________________

Source: Burns and Mitchell

[START_REF] Burns | Measuring Business Cycles[END_REF]

.

Table III .

 III The Ratio of the Estimated Output-Labor to Output-Capital Gains, U.S., 1948-83 a

	_________________________________________________________________________
	ω 0.000 _________________________________________________________________________ 2π/ω in G YL (ω) G YK (ω) G YL (ω)/G YK (ω) 0.088 0.262 0.338 ∞ Short-Run Short-Run Short-Run 1.00 4.00 0.80 1.00
	0.025 0.80 0.80	256.456			0.091		0.294		0.309
	0.049	2.00 0.60	127.967			0.098		0.367		0.268
	0.074		85.369			0.143		0.367		0.389
	0.098 0.60 0.60	63.984			0.203		0.316		0.643
	0.123	0.00 0.40	51.208			0.280		0.342		0.820
	0.147 0.40 0.40	42.656			0.392		0.287		1.365
	0.172		36.573			0.387		0.281		1.375
	0.196 -2.00 0.20 0.221 0.20 0.20	32.008 28.444	0.471 0.504 95% Critical Value 95% Critical Value		0.280 0.261		1.680 1.931
	0.245		25.604			0.538		0.256		2.106
	0.270 0.00 0.000 -4.00 0.000 0.00 0.000 0.00 0.000	23.271 0.524 0.524 0.524 0.524		1.048 1.048 1.048 1.048	0.543 1.572 1.572 1.572 1.572	0.237 2.096 2.096 2.096 2.096	2.620 2.620 2.620 2.620	2.293 3.144 ω 3.144 ω 3.144 ω 3.144
	0.294	∞ ∞ ∞	21.335 12 12 24 24 24 12	8 8 8	6 6 6	0.575 4 4 4	3 3 3	0.224		2.570 2 ξ 2 ξ 2 ξ
	0.319		19.690			0.579		0.211		2.749
	0.344 Figure 3. Gain of Output and Capital, U.S., 1948-83 18.286 0.586 0.219		2.672
	0.368		17.065			0.581		0.220		2.637
	0.393		16.000			0.600		0.218		2.749
	0.417		15.060			0.583		0.205		2.839
	0.442		14.222			0.566		0.200		2.827
	0.466		13.475			0.561		0.220		2.542
	0.491		12.799			0.542		0.218		2.486
	0.515		12.191			0.557		0.210		2.655
	_________________________________________________________________________

a. ω denotes the frequency, 2π /ω denotes the periodicity in quarters, and G YL (ω) and G YK (ω) denote output-labor and output-capital gain functions, respectively. Figure 1. Coherence of Output and Capital, U.S., 1948-83 Figure 2. Phase of Output and Capital, U.S., 1948-83

Footnotes

* I am grateful to Shomu Banerjee, Bob Carpenter, Paul Romer, Mark Watson, and an anonymous referee for reading an earlier version of this manuscript and providing detailed comments. I also thank Martin J. Bailey, Mark Bergen, Hashem Dezhbakhsh, Hal Fried, Maxwell Fry, Amihai Glazer, Jinook Jeong, Roger Kormendi, David Lilien, Giovanna Mossetti, Will Roberds, and Shelton Schmidt as well as the economics seminar participants at Emory University for helpful discussions and suggestions. Comments received from Mark Bergen, Kevin Denny, and David Smyth on my 1990 Economics Letters paper led me to the ideas presented here. All errors are mine.

1. Statistically, the increasing importance of the capital stock in the growth of the U.S. economy has been documented in other studies as well, but the figures are not as high as Romer's estimates. For example, the figures cited in Maddison [START_REF] Maddison | Growth and Slowdown in Advanced Capitalist Economies: Techniques of Quantitative Assessment[END_REF] for the weight of the capital stock are in the range 0.21-0.40. The estimates of the post-war capital elasticity of the output reported by Levy [START_REF] Levy | Aggregate Output, Capital, and Labor in the Post-War U.S. Economy[END_REF] are in the range 0.44-0.55. However, these figures were estimated in studies that cover much shorter periods than Romer [START_REF] Romer | Crazy Explanations for the Productivity Slowdown[END_REF].

2. Although it seems that Marshall [START_REF] Marshall | The Principles of Economics[END_REF] was the first to explicitly argue that the stock of capital is fixed in the short-run, the idea itself can be traced back to Smith and Ricardo. Marshall assumed that the industry in the short period can be treated as if it were in static equilibrium, which together with the assumption of fixed equipment stock, resembles the agricultural sector model of Ricardo. Marshall's fixed capital plays the same role as Ricardo's land, 'original' and 'indestructible' within the period. Marshall treated his short period as a single period in the manner of Smith or Ricardo and invoked the constancy (or approximate constancy) of the capital stock of the industry as a justification for treating the single period as self-contained [START_REF] Hicks | Capital and Growth[END_REF].

According to Robinson [START_REF] Robinson | The Production Function and the Theory of Capital[END_REF], Keynes treated the existing stock of capital in the short-run simply as a "part of the environment in which labour works."


Daniel Levy

Investment-Saving Comovement under Endogenous Fiscal Policy by

Keywords: JEL Classification: F41, E62, H39 Feldstein-Horioka Puzzle, Investment, Saving, Capital Mobility, Endogenous Fiscal Policy

model by including flexible exchange rate and by introducing endogenous fiscal policy. Using this model, I demonstrate how a positive investmentsaving correlation can arise in a world with endogenous fiscal policy. I show that this correlation does not depend on capital mobility and therefore is compatible with any degree of capital mobility. This implies that the observed investment-saving comovement is not necessarily due to imperfect capital mobility. The model has a testable implication: it predicts a lack of Granger causality from private saving to private investment. Empirical examination of this prediction indicates that U.S. time series data is compatible with the hypothesis of endogenous fiscal policy during a flexible exchange rate period, but not during a fixed exchange rate period.

1

In a closed economy, private investment can only be financed by domestic saving and therefore investment and saving will be correlated. However, in an open economy, at least part of the investment can be financed by foreign saving. Therefore, in an open economy, national saving and investment may in principle diverge and move independently of each other.

However, the empirical findings by [START_REF] Feldstein | Domestic Saving and International Capital Flows[END_REF] and [START_REF] Feldstein | Domestic Saving and International Capital Movement in the Long Run and the Short Run[END_REF] indicate that investment and saving are highly correlated across countries. They interpret these findings as evidence of a lack of perfect capital mobility. 1 This interpretation of the investment-saving comovement has been very controversial and a significant number of studies disagree with it 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. 2 Several authors, like [START_REF] Tobin | Domestic Saving and International Capital Movements in the Long-Run and the Short-Run" by M. Feldstein: Comment[END_REF], [START_REF] Westphal | Domestic Saving and International Capital Movements in the Long-Run and the Short-Run" by M. Feldstein: Comment[END_REF], [START_REF] Roubini | Current Account and Budget Deficits in an Intertemporal Model of Consumption and Tax Smoothing[END_REF], [START_REF] Summers | Tax Policy and International Competitiveness[END_REF], have offered an alternative explanation for the investment-saving comovement. If the government is concerned about balancing a current account deficit, it can set the size of its spending (or its budget deficit) endogenously by reacting to the gap between private saving and investment. If the government uses such an endogenous fiscal policy, then private investment and national saving will also be correlated; i.e., this will be true regardless of the degree of capital mobility.

From a theoretical point of view, standard macroeconomic theory assigns the government budget an exogenous policy role. However, its discretionary use by governments for stabilization purposes implies that policy makers actually react to events and situations occurring in the economy. This makes fiscal policy endogenous. In the new classical framework fiscal policy is treated in a similar manner except that the emphasis is on unanticipated implementation of the policy. 3 Whether the government's budget policy is exogenous or endogenous is important because of its policy implications in the context of investment-saving correlation: Feldstein and Horioka's finding that variations in saving are associated with variations in investment, implies that under exogenous fiscal policy a reduction in the U.S. budget deficit will not improve external balance, rather it will increase investments. This conclusion, however, is not necessarily correct if fiscal policy is endogenous in the sense described above. 4 In this paper I present a variant of a standard neoclassical model modified for a large open economy with flexible exchange rates. Using the model, I analyze the implications of perfect capital mobility and immobility for the investment-saving relationship and the crowding out effects of expansionary fiscal policy on investment and trade balance.

The model presented here is an extension of the model discussed in [START_REF] Feldstein | Domestic Saving and International Capital Movement in the Long Run and the Short Run[END_REF]. The present model expands beyond Feldstein's model by explicitly introducing the market for foreign exchange as well as by addressing the role of fiscal policy in the context of investment-saving comovement. Unlike [START_REF] Feldstein | Domestic Saving and International Capital Movement in the Long Run and the Short Run[END_REF], who implicitly assumes that fiscal policy is exogenous, I allow for the possibility of endogeneity in the conduct of fiscal policy.

The model demonstrates how a positive investment-saving correlation can arise in a world with endogenous fiscal policy. Since this correlation does not depend on capital mobility and is therefore compatible with any degree of capital mobility, I conclude that tests of capital mobility that are based on investment-saving correlation are not likely to be informative about the true degree of capital mobility.

I also discuss testable implications of the model. In particular, I show that an endogenous fiscal policy hypothesis of the type considered in this paper predicts a lack of Granger causality from private saving to private investment. Empirical examination of this prediction indicates that the U.S. time series data is compatible with the hypothesis of endogenous fiscal policy during a flexible exchange rate period, but not during a fixed exchange rate period. The implication is that the observed investment-saving correlation in the U.S. time series may indeed be a result of an endogenous fiscal policy.

The paper is organized as follows. Section 1 describes the structure of the model. In Section 2 the implications of the model are analysed under the assumption of exogenous fiscal policy. In Section 3, the implications of the model are analysed under the assumption of endogenous fiscal policy. In Section 4, testable implications of the model are empirically examined using the U.S. time series data. The paper ends with conclusions.

The Model

Following [START_REF] Feldstein | Domestic Saving and International Capital Movement in the Long Run and the Short Run[END_REF], assume that the demand for gross domestic investment, I, is a decreasing function of real interest rate, r:

I = I(r) + ξ I I′ <0 (1)
where ξ I is a zero mean random term with constant variance. Supply of private saving, PS, is an increasing function of the real interest rate:

PS = S(r) + ξ s S′ >0 (2)
where ξ s is a zero mean random term with constant variance. I assume that the correlation between ξ I and ξ s is zero. I make this assumption in order to focus on the role of fiscal policy in generating investment-saving comovement. Otherwise we need to address a possibility of a "common cause" mechanism, which is beyond the scope of this paper.

Exports, X , and imports, M, depend on the real exchange rate, R, and are given by ( 3) and ( 4):

X = X(R) + ξ X X′ > 0 (3) M = M(R) + ξ M M′ < 0 (4)
where ξ X , and ξ M are random shocks with zero mean and constant variance. From national income accounting, net exports, NX, equal net capital outflow (net foreign investment), F, which in turn is defined as capital outflow, F o , minus capital inflow, F i , and is a decreasing function of the real interest rate. Thus,

F = F o -F i = F(r) + ξ F F′≤ 0 (5)
where ξ F is a zero mean random term with constant variance. In case of perfect capital mobility, F′ = -∞, and if capital is completely immobile, then F′ = 0.

Exogenous Fiscal Policy

Now assume that the government's receipts are fixed at T 0 , and that the expenditures, G, are set exogenously. The budget deficit is given by

G -T = G 0 + ξ G -T 0 (6)
where ξ G is a fiscal policy shock. The model is closed by the national income identity

PS -(G -T) -I = X -M (7)
which states that the excess of national saving over investment must match the trade balance.

Since F = X -M, (3), (4), and (5) imply that

F(r) + ξ F = X(R) + ξ X -M(R) -ξ M (8)
which after total differentiation and rearrangement, yields

dR = F′dr -dξ X + dξ M + dξ F X′ -M′ . ( 9
)
Since X′ > 0, and M′ < 0, the denominator in ( 9) is always positive. Therefore, equation ( 9) implies that

dR dr = F′ X′ -M′ ≤ 0.
That is, unless capital is completely immobile, an increase in interest rate (dr > 0) will lead to a lower exchange rate since the resulting capital inflow will cause an appreciation of the domestic currency. It also implies that a positive (negative) shock to imports (exports) devalues domestic currency since it results in higher (lower) demand for (supply of) foreign currency.

Now substitute (1), ( 2), ( 3), (4), and ( 6) into (7) to get

S(r) + ξ s -G 0 -ξ G + T 0 -I(r) -ξ I = X(R) + ξ X -M(R) -ξ M . ( 10
)
Taking the total differential of (10), and using (9) for dR, yields

dr = dξ I + dξ G -dξ s + dξ F S′ -I′ -F′ . ( 11
)
Since S′ >0, I′ <0, and F′≤ 0, the denominator in ( 11) is always positive. The equation states that as long as capital is not perfectly mobile (i.e., F′ > -∞), the interest rate rises when there is a positive shock to either investment (dξ I > 0), government expenditures (dξ G > 0), or demand for net foreign investment (dξ F > 0). However, if capital is perfectly mobile (i.e., F′ = -∞),

then either shock will instantaneously result in capital inflow, completely satisfying the excess demand for resources, and thus leaving the interest rate unchanged.

Substituting (11) into (9), gives dR in terms of all the possible shocks of the model:

dR = F′ dξ I + dξ G -dξ s + dξ F (X′ -M′)(S′ -I′ -F′) + dξM -dξ X + dξ F X′ -M′ . ( 12
)
Note that both denominators in equation ( 12) are positive. Taking the total differential of (5) and combining it with (11) yields

dF = F′dr + dξ F = F′ dξ I + dξ G -dξ s + dξ F S′ -I′ -F′ + dξ F (13)
which shows the response of net capital outflow ( or net foreign investment), and thus of trade balance, to various shocks. Equations ( 12) and ( 13) can be used jointly for comparative statics under alternative hypotheses about capital mobility. For example, with perfect capital mobility (i.e., F′ = -∞), an increase in government spending (dξ G > 0) causes a decrease in net foreign investment (F) "one for one" and therefore results in capital inflow since lim

F ′→-∞ dF dξ G = -1. Simultaneously, lim F ′→-∞ dR dξ G = - 1 X′ -M′ < 0,
which means that domestic currency appreciates reducing the external balance. In other words, an expansionary fiscal policy completely crowds out net exports, but has no effect on the interest rate as lim

F ′→-∞ dr dξ G = 0.
At the other extreme, if capital is completely immobile ( F′ = 0), then the increased government demand must be satisfied by domestic resources. The first term on the right hand side of ( 12) and ( 13) vanishes when F′ = 0. Therefore, an expansionary fiscal policy leaves the current account and the exchange rate unchanged and instead increases the interest rate since lim

F ′→0 dr dξ G = 1 S′ -I′ >0.
This leads to a crowding out of investment because lim

F ′→0 dI dξ G = I′ S′ -I′ <0,
as can be seen from equation ( 14) below. Using a similar argument, it follows that under partial capital mobility, (i.e., 0 > F′ > -∞), a higher budget deficit will crowd out both investment as well as net export.

In order to see the relationship between investment and national saving, take the total derivative of (1) and use (11) to get

dI = I′dr + dξ I = I′ dξ I + dξ G -dξ s + dξ F S′ -I′ -F′ + dξ I (14)
which means that

dI dξ s = - I′ S′ -I′ -F′ . ( 15
)
Under perfect capital mobility, lim

F ′→-∞ dI dξ s =0,
and so any change in domestic private saving (which changes national saving by an equal amount since government saving is exogenous) is matched by a corresponding change in capital flow, leaving investment unchanged. Thus under perfect capital mobility, investment and national saving will not be correlated. If capital is immobile, however, i.e., if F′ = 0, then lim

F ′→0 dI dξ s = -I′ S′ -I′
and therefore 0 < dI dξ s ≤ 1. That is, higher national saving will lead to higher investment. It is this result that led [START_REF] Feldstein | Domestic Saving and International Capital Flows[END_REF] to interpret the positive investment-saving relationship they found as an indicator of a lack of perfect capital mobility.

Endogenous Fiscal Policy

Now consider a situation where the government has a reaction function through which it responds to current account imbalances by adjusting its spending according to the gap between private saving and investment. In particular, suppose that fiscal policy reaction function is given by

G -T = α(PS -I) + ξ G -T 0 ( 16
)
where 0 < α ≤ 1. Substituting (1), ( 2), ( 3), (4), and ( 16) into (7), taking its total derivative, and using (9) for dR yields

dr = (1 -α)(dξ s -dξ I ) -dξ F -dξ G (1 -α)(I′ -S′) + F′ . ( 17
)
Here, if the government follows an endogenous fiscal policy of the type assumed in ( 16), then investment and national saving will be correlated even if capital is perfectly mobile. Therefore, the observed positive correlation between investment and private saving is not necessarily an indicator of a lack of perfect capital mobility. To see this, suppose that there is a positive change in private saving. i.e., dξ s > 0. Such a change will be followed by an adjustment of the government's budget deficit according to (16). To get the exact expression, take the total derivative of ( 16), replace dr by ( 17), and find d(G -T). This yields

d(G -T) = α(S′ -I′)dr + αdξ s -αdξ I + dξ G = α(S′ -I′) (1 -α)(dξ s -dξ I ) -dξ F -dξ G (1 -α)(I′ -S′) + F′ + αdξ s -αdξ I + dξ G . ( 18
)
When capital is perfectly mobile, i.e., F′ = -∞, the first term on the right hand side of ( 18)

vanishes and

d(G -T ) dξ s = α.
The total differential of national saving is given by

dNS = (1-α)S′ + α I′ dr + (1 -α)dξ s + αdξ I -dξ G , (19)
and since dr = 0 under perfect capital mobility, national saving will respond to the change in private saving by dNS dξ s = (1 -α). Note that in such an environment, private investment does not respond to changes in private saving. To see this, derive dI dξ s by taking the total differential of (1) and substituting (17) for dr:

dI dξ s = I′(1 -a) (1 -α)(I′ -S′) + F′ . ( 20
)
Under perfect capital mobility we get lim

F ′→-∞ dI dξ s =0.
Thus, this model makes a testable prediction: it predicts a lack of Granger causality from private saving to investment. However, national saving responds to changes in investment since dNS dξ I = α, which does not depend on F′. Hence, in the case of endogenous fiscal policy, investment and national saving will be correlated no matter what the degree of capital mobility is.

Note that the crowding out effect of fiscal policy here will be similar to the effect when fiscal policy is exogenous. To see this, substitute ( 17) into (9) to get

dR = F′ (1 -α)(dξ s -dξ I ) -dξ F -dξ G (X′ -M′) (1 -α)(I′ -S′) + F′ + dξM -dξ X + dξ F X′ -M′ (21)
which implies that under perfect capital mobility lim

F ′→-∞ dR dξ G = lim F′ → -∞ - F′ (X' -M')[(1 -a)(I' -S') + F']
< 0 by L'Hôspital's rule. Therefore, expansionary fiscal policy leads to an appreciation of domestic currency and consequently causes a crowding out of net exports. Since under perfect capital mobility dr = 0 from ( 17), dI dξ G =0 and fiscal policy will have no effect on investment.

However, if capital is immobile, then dR dξ G =0, while

dI dξ G =- I′ (1 -α)(I′ -S′)
< 0, so we get a crowding out of investment.

In summarize, the model presented here shows that a positive correlation between domestic private investment and national saving is compatible with both, exogenous as well as endogenous fiscal policy. In case of exogenous fiscal policy, under perfect capital mobility, domestic investment and national saving are not correlated, while under immobile capital they tend to move together. If fiscal policy is endogenous, then domestic investment and national saving will be correlated even if capital is perfectly mobile. In such case, an observed strong correlation between national saving and domestic investment will not necessarily indicate a lack of perfect capital mobility. This hypothesis has a testable implication: it predicts a lack of Granger causality from private saving to investment.

Granger Causality and Cointegration Test Results

As was shown in Section 4, the hypothesis of endogenous fiscal policy of the type assumed in this paper predicts a lack of Granger causality from private saving to investment. As a test of this implication I run a bivariate Granger causality test between time series of gross private saving and gross private domestic investment using U.S. annual data covering the period 1929-89. The data for the 1929-82 period is taken from the NIPA tables of the BEA. For the 1983-89 period the data come from the Economic Report of the President (1990). The series are measured in 1982 dollars. The plots of the time series of domestic investment and private saving are shown on Figure 1.

The Granger causality test equation is given by

I t = α + β n I t -n Σ n =1 4 + γ n PS t -n Σ n =1 4 + ε t , ( 22
)
where I t denotes gross private domestic investment and PS t denotes gross private saving. The null hypothesis is that private investment is not Granger-caused by private saving. Technically, this is tested by examining whether or not γ n = 0 for each n simultaneously.

The results are reported in Table 1. I run the causality test for the entire sample period as well as for two sub-periods, which correspond to periods of fixed and flexible exchange rate regimes, respectively. According to the results, the null hypothesis cannot be rejected for the entire sample period. The null hypothesis cannot be rejected also for the second subperiod, but it is rejected at the 5% level of significance for the first subperiod. This means that the data is compatible with the hypothesis of endogenous fiscal policy during a flexible exchange rate period, but not during a fixed exchange rate period. Obviously, these findings do not prove that fiscal policy was endogenous over 1973-89 period because a lack of Granger causality from private saving to private investment could in principle be a consequence of other types of policies. For comparison purposes it should be mentioned that [START_REF] Summers | Tax Policy and International Competitiveness[END_REF] and [START_REF] Bayoumi | Saving-Investment Correlations: Immobile Capital, Government Policy, or Endogenous Behavior?[END_REF], using a cross-section data of OECD and ten industrial countries, respectively, find that an endogenous fiscal policy of the type modelled here might be playing an important role in generating the positive investment-saving comevement obsereved in the data of open economies.

It is possible to verify these Granger causality test results by considering a cointegration relationship between domestic investment and private saving. 5 According to the Error-Correction Representation Theorem proved by [START_REF] Engle | Cointegration and Error-Correction: Representation, Estimation, and Testing[END_REF], the concepts of cointegration and Granger causality are directly related. Specifically, they show that if two series are cointegrated, then there should be a Granger causality between these series at least in one direction. Thus, by examining the cointegration property of the investment-saving comovement, we can check whether the Granger-causality test results are consistent with cointegration tests.

Several authors have examined the cointegration relationship between investment and saving using the U.S. time series data. See for example [START_REF] Miller | Are Saving and Investment Cointegrated?[END_REF], [START_REF] Otto | National Saving and Domestic Investment in the Long Run: Some Time Series Evidence for the U.S. and Canada[END_REF], [START_REF] Leachman | Saving, Investment, and Capital Mobility Among OECD Countries[END_REF], [START_REF] Gulley | Are Saving and Investment Cointegrated? Another Look at the Data[END_REF], and [START_REF] Levy | Is Feldstein-Horioka Puzzle Really a Puzzle?[END_REF]. All these authors, however, focus their attention on the relationship between the time series of national saving and domestic investment. Here I am interested in examining the relationship between private saving and domestic investment. 6 To test for cointegration, we first need to establish unit root properties of the time series of private saving and domestic investment. To determine whether private saving and domestic investment series are stationary, we examine each series for a presence of unit roots using the Augmented Dickey-Fuller tests of the form

∆ x t = α 0 + α 1 t + γ x t -1 + φ i ∆ x t -i Σ i =1 4 + ε t , (23)
where x t is the series we are examining and t is a linear time trend. The null hypothesis is that γ = 0, which means that the series contains an unit root and is thus nonstationary, i.e., x t ~I(1).

The alternative hypothesis is that the series are stationary, that is, x t ~I(0).

Initially, we test the hypothesis of an unit root in the series measured in levels. The results are reported in the first column of Table 2. As the values of the Augmented Dickey-Fuller tstatistics indicate, the hypothesis of nonstationarity cannot be rejected for domestic investment (denoted I) and private saving (denoted PS). Next, we test whether the first differences of the series are nonstationary. The figures presented in the last column of Table 2 indicate that the hypothesis of an unit root in the differenced series can be rejected. 7 Therefore, we conclude that private saving and domestic investment series are I(1), and thus can be represented as difference stationary processes. The results reported by [START_REF] Miller | Are Saving and Investment Cointegrated?[END_REF], [START_REF] Otto | National Saving and Domestic Investment in the Long Run: Some Time Series Evidence for the U.S. and Canada[END_REF], [START_REF] Leachman | Saving, Investment, and Capital Mobility Among OECD Countries[END_REF], [START_REF] Gulley | Are Saving and Investment Cointegrated? Another Look at the Data[END_REF], and [START_REF] Levy | Is Feldstein-Horioka Puzzle Really a Puzzle?[END_REF] on the unit root properties of domestic investment in the U.S. are essentially identical to what I report here.

To test for cointegration between the time series of private saving and domestic investment, I use [START_REF] Johansen | Statistical Analysis of Cointegration Vectors[END_REF] maximum likelihood method which seems to have better statistical properties than the more commonly used [START_REF] Engle | Cointegration and Error-Correction: Representation, Estimation, and Testing[END_REF] two-step method. Below I define the test statistics used in this approach. For a detailed description of Johansen's maximum likelihood method, see Johansen's original article or [START_REF] Banerjee | Cointegration, Error-Correction, and the Econometric Analysis of Non-Stationary Data[END_REF]. [START_REF] Johansen | Statistical Analysis of Cointegration Vectors[END_REF] proposes two tests for estimating the number of cointegrating vectors.

The first, called maximal eigenvalue test, is given by the statistic

λ max = -n log 1 -λ r ( 24
)
where n is the number of observations, and λ r is the r-th eigenvalue determined by solving the determinantal equation associated with the residual product moment matrix constructed using the residuals' matrices. The maximal eigenvalue test is designed to test H(r -1) against H(r). That is, the null hypothesis is that there are (r -1) cointegrating vectors against the alternative of r.

The second test, called trace test, is designed for testing H(r) against H(m), where r < m.

The trace statistic is given by

J T =-n log 1 -λ i Σ i = r + 1 m . ( 25
)
The cointegration test results are presented in Table 3. The critical values I use are taken from Osterwald-Lenum (1992). 8 In estimating the cointegration vectors I use VAR(4).

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. 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 . Full rank, that is r = m, only indicates that the process is stationary.

Following this sequential procedure, we find that the time series of domestic investment and private saving are cointegrated only during the 1929-72 period (with one cointegrating vector, i.e., r = 1). To see this, look at the middle part in Table 3: the values of both test statistics, λ max = 12.79 and J T =13.40, are slightly bigger than the corresponding 10% critical values. Therefore, the null of no cointegrating vectors ( r = 0) is rejected in favour of the alternative with 10% significance. Now we proceed to the next hypothesis, r ≤ 1. Here we see that both test statistics (0.31 and 0.31) are below the corresponding critical values. 9 Therefore the null of one or less cointegrating vectors, ( r ≤ 1), cannot be rejected. Since the hypothesis of zero cointegrating vector was rejected in the preceding stage, we conclude that r = 1. That is, domestic investment and private saving are cointegrated with one cointegrating vector.

During the sample periods 1929-89 and 1973-89, the test statistics are not statistically significant in the first stage of the testing. Therefore the null of zero cointegrating vectors ( r = 0) cannot be rejected for these periods. That is, domestic investment and private saving are not cointegrated for these sample periods. Thus, we find that the series are cointegrated during fixed exchange rate period, but not during flexible exchange rate period. [START_REF] Miller | Are Saving and Investment Cointegrated?[END_REF] reports similar findings for the time series of national saving and domestic investment.

These findings are consistent with Granger causality test results reported above. This is because a cointegration relationship between the two series implies Granger causality at least in one direction. Indeed, for the 1929-72 period where we find an evidence of cointegration, we rejected the null of no Granger causality from private saving to domestic investment. For the other two sample periods, 1929-89 and 1973-89, we find that the two series are not cointegrated. And the finding that there is no Granger causality during these sample periods is clearly consistent with the cointegration test results. 10

An additional implication of the observed cointegration of domestic investment and private saving during the fixed exchange rate period can be derived by looking at the national income accounting identity given in equation ( 7), which can be rewritten as

PS -I = X -M + G -T . ( 26
)
If PS and I are cointegrated, then their linear combination given on the left hand side of equation ( 26) should be stationary. 11 Consequently, the right hand side of ( 26), which is a sum of budget deficit and current account balance, should also be stationary during the fixed exchange rate period. On the other hand, during the periods of no cointegration, 1929-89 and 1973-89, we expect this sum to be nonstationary.

To examine the empirical validity of this prediction, I have applied Augmented Dickey-Fuller unit root test to the X -M + G -T series. The results reported in Table 2 indicate that the unit root tests are not fully consistent with the cointegration test results. During the 1929-89 and 1973-89 periods the X -M + G -T series are nonstationary (the null of unit root cannot be rejected) as predicted by cointegration test results. However, it is nonstationary also during the 1929-72 period which is inconsistent with the finding that domestic investment and private saving are cointegrated during that period.

Conclusions

In this paper I expand the standard neoclassical open economy model by introducing endogenous fiscal policy. Using the model, I demonstrate analytically how a positive investment-saving correlation can arise in a world with endogenous fiscal policy. I also show that this correlation does not depend on capital mobility and therefore is compatible with any degree of capital mobility. This means that the observed investment-saving comovement in the U.S. time series data does not necessarily imply imperfect capital mobility. The hypothesis of endogenous fiscal policy predicts a lack of Granger causality from private saving to investment.

Empirical examination of this prediction indicates that the U.S. time series data of private saving and investment is compatible with the hypothesis of endogenous fiscal policy during a flexible exchange rate period, but not during a fixed exchange rate period.

The implication is that the investment-saving comovement observed in the U.S. time series data may indeed be due to an endogenous fiscal policy, where the government is trying to balance the current account by setting its budget plans according to the gap between private saving and investment.

How can we explain the differences in results between the fixed and flexible exchange rate periods? One possibility is that during fixed exchange rate period investment and saving were cointegrated simply because capital was not very mobile. The abandonment of the Breton Wood system and the elimination of capital controls in the early 1970s along with the increased integration of worlds financial markets during the last 20 years may have led to an increased international capital mobility which eliminated the long-run (i.e., cointegration) relationship between investment and saving.

An alternative explanation may be given along the lines suggested in this paper. That is, it is possible that the government's fiscal policy was indeed endogenous during the flexible exchange rate period, but not during fixed exchange rate period. Note that in the context of the model presented in the paper, what I have in mind is a very specific form of endogeneity: the government reacts to current account imbalances by adjusting its budget balance according to the gap between private saving and investment.

In general it may be argued that this kind of policy is more plausible during fixed, rather than flexible exchange rate periods as [START_REF] Miller | Are Saving and Investment Cointegrated?[END_REF] suggests. An examination of the U.S. trade deficit development over the last 10-15 years, however, suggests the opposite. The continuous appreciation of the dollar during the 1980s has led to a large trade and current account deficits which was viewed by many observers and administration officials as a cause for great concern.

The increase in trade deficit led to a loss of exports and substitution of imports for domestic production. Consequently, there was a significant pressure on policy makers from the affected industries to reduce the trade deficit. Therefore, an endogenous fiscal policy was a rather likely scenario for the U.S. during the flexible exchange rate period. 12 Summers (1988) and [START_REF] Bayoumi | Saving-Investment Correlations: Immobile Capital, Government Policy, or Endogenous Behavior?[END_REF] reach a similar conclusion using cross-country data, which includes the U.S.

An extension of this study tries to evaluate the empirical validity of the endogenous fiscal policy hypothesis by directly estimating a fiscal policy reaction function of the type considered in this paper using U.S. time series data. Although preliminary results suggest that the hypothesis of endogenous fiscal policy cannot be rejected, this question is still under investigation.

Notes

1. Studies that expand Feldstein and Horioka's analysis to different time periods and samples, include [START_REF] Caprio | Domestic Saving, Current Accounts, and International Capital Mobility[END_REF], [START_REF] Murphy | Capital Mobility and Relationship between Saving and Interest Rates in OECD Countries[END_REF], Obstfeld (1986a[START_REF] Obstfeld | How Integrated Are World Capital Markets: Some New Tests[END_REF][START_REF] Wong | What Do Saving-Investment Relationships Tell Us about Capital Mobility?[END_REF].

2. See, e.g., [START_REF] Tobin | Domestic Saving and International Capital Movements in the Long-Run and the Short-Run" by M. Feldstein: Comment[END_REF], [START_REF] Westphal | Domestic Saving and International Capital Movements in the Long-Run and the Short-Run" by M. Feldstein: Comment[END_REF], Obstfeld (1986a[START_REF] Obstfeld | How Integrated Are World Capital Markets: Some New Tests[END_REF], [START_REF] Frankel | Political vs. Currency Premia in International Real Interest Rate Differentials[END_REF], [START_REF] Ghosh | International Capital Mobility and Optimal Current Account Behavior: An Empirical Investigation[END_REF], [START_REF] Tesar | Saving, Investment, and International Capital Flows[END_REF], and [START_REF] Baxter | Explaining Saving-Investment Correlations[END_REF].

3. These studies often assume fiscal dominance, which means that fiscal authorities set their budget policy exogenously, and monetary authorities then have no choice but accommodate it. See, e.g., [START_REF] Sargent | Some Unpleasant Monetarist Arithmetics[END_REF]. From the empirical point of view, classification of government budget policy as exogenous or endogenous seems largely to depend on the hypothesis been tested. For example, [START_REF] Evans | Interest Rates and Expected Future Budget Deficits in the United States[END_REF], [START_REF] Ghosh | International Capital Mobility and Optimal Current Account Behavior: An Empirical Investigation[END_REF][START_REF] Moreno | Saving, Investment, and the U.S. External Balance[END_REF] model government budget deficit as exogenous while [START_REF] Dooley | International Capital Mobility: What Do Investment-Saving Correlations Tell Us?[END_REF], [START_REF] Fieleke | National Saving and International Investment[END_REF], and [START_REF] Pigott | Economic Consequences of Continued U.S. External Deficits[END_REF] treat the budget policy as endogenous.

4. The "rule of thumb" published by Congressional Budget Office (1985, p. 75) suggests that government budget policy is endogenous in CBO's view.

5. I thank the anonymous referee for suggesting this idea.

6. Although [START_REF] Bayoumi | Saving-Investment Correlations: Immobile Capital, Government Policy, or Endogenous Behavior?[END_REF] examines the relationship between private saving and domestic investment, he does not use the cointegration methodology since he estimates cross-section regressions using data from ten industrial countries. [START_REF] Argimón | Saving, Investment, and International Capital Mobility in EC Countries[END_REF] do examine the cointegration relationship between private saving and domestic investment but their data set does not include the U.S.

7. The unit root test results for the X -M + G -T series are discussed below.

8. I choose not to use more commonly used critical values tabulated by [START_REF] Johansen | Maximum Likelihood Estimation and Inference on Cointegration with Applications to the Demand for Money[END_REF] since [START_REF] Podivinsky | Testing Misspecified Cointegrating Relationships[END_REF] shows that those critical values may be invalid for sample sizes of 100 or less.

9. 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.

10.However, it should be noted that an opposite result on Granger causality during these sample periods would not contradict cointegration test results either. This is because if there is no cointegration, then either one, Granger causality and Granger noncausality, are possible.

11.Actually, this analysis is based on the finding (not reported in Table 3) that the cointegrating vector is [1,-1]. [START_REF] Leachman | Saving, Investment, and Capital Mobility Among OECD Countries[END_REF] reports a similar finding.

12.It should be mentioned, however, that setting the budget deficit according to the gap between private saving and investment is not the only way for reducing current account deficit. A reduction in a current account deficit can be accomplished also by means of monetary policy by affecting the exchange rate and thus the level of economic activity. Other possibilities include the use of investment tax credit, the adjustment of depreciation allowance schemes, etc. See [START_REF] Martin | National Saving and International Investment[END_REF]. JT is the trace test statistic defined in (25). r is the number of cointegrating vectors.

Figure 1 .

 1 Figure 1. Private Saving and Domestic Investment, U.S., 1929-89, Billions of 1982 Dollars.(Source: National Income and Product Accounts of the U. S., Bureau of Economic Analysis, Department of Commerce, 1992.)

  a X -M + G -T -2.04 -3.62 c Note: I used Box-Pierce, Ljung-Box, and Lagrange Multiplier tests to verify that the error terms in all the regressions reported in this paper are not serially correlated. Superscripts a, b, and c indicate statistical significance at 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% significance critical values for the Augmented Dickey-Fuller t-statistics are as follows: -4.12, -3.49, and -3.17 for the period 1929-89; -4.20, -3.52, and -3.19 for the period 1929-72; and -4.61, -3.71, and -3.29 for the period 1973-89, respectively. These critical values are tabulated in MacKinnon (1991). Notation: I = Domestic Investment, PS = Private Saving, X = Exports, M = Imports, G = Government Expenditure, and T = Government Revenue.

Table 1 .

 1 Granger Causality Test Results : The H 0 hypothesis is that private investment is not Granger caused by private saving. Superscript b indicates a 5% significance.

	Sample Period	F-Statistic	p-value	Is the H 0 rejected?
	1929-89	1.93	0.12	No
	1929-72	2.95 b	0.03	Yes
	1973-89	0.47	0.75	No

Note

Table 2 .

 2 Unit Root Test Results

	Period	Series	Augmented Dickey-Fuller t-statistic
			__________________________________
			Levels	First differences
	1929-89	I	-2.46	-4.25 a
		PS	-2.23	-4.

Table 3 .

 3 Cointegration Test Results

	Sample	Variables	Test	H 0	H 1	Test	Critical Values
	Period	Included				Statistic	95%	90%
	1929-89	I, PS	λ max	r = 0 r ≤ 1	r = 1 r = 2	10.64 0.24	14.06 3.76	12.07 2.68
			J T	r = 0 r ≤ 1	r ≥ 1 r = 2	10.88 0.24	15.41 3.76	13.32 2.68
	1929-72	I, PS	λ max	r = 0 r ≤ 1	r = 1 r = 2	12.79 c 0.31	14.06 3.76	12.07 2.68
			J T	r = 0 r ≤ 1	r ≥ 1 r = 2	13.40 c 0.31	15.41 3.76	13.32 2.68
	1973-89	I, PS	λ max	r = 0 r ≤ 1	r = 1 r = 2	8.98 4.42	14.06 3.76	12.07 2.68
			J T	r = 0 r ≤ 1	r ≥ 1 r = 2	13.21 4.42	15.41 3.76	13.32 2.68

Note: Superscript c indicates a 10% statistical significance.

The critical values are taken from Osterwald-Lenum (1992). λ max is the maximal eigenvalue test statistic defined in (24).

Acknowledgements

I thank the anonymous referee, the editor, and an associate editor of this journal for helpful comments and suggestions. This paper is an extension of Chapter 1 of my Ph.D. dissertation. I thank my Ph.D. thesis committee members, Maxwell Fry, Jack Johnston, David Lilien, and Giovanna Mossetti for commenting on earlier drafts of this paper.

This paper is based on the University of California/Irvine Economics Working Paper No. 90-91-04 titled "Investment-Saving Comovement, Capital Mobility, and Fiscal Policy," which was presented at the 1991 National Bureau of Economic Research Conference on the Macroeconomic Effects of Fiscal Policy. I benefited from the comments made by the participants of that conference, including Andrew Abel, V.V. Chari, Robert Eisner, Robert E. Lucas, Jr., Assaf Razin, and especially by my discussants, Marianne Baxter and Maurice Obstfeld. An earlier version of this paper was also presented at Bar Ilan, Ben Gurion, Board of Governors, Emory, Indiana (Bloomington and Indianapolis), Louisiana State University, Northeastern, State University of New York at Albany, Tulane, Union, and University of California at Irvine. I am grateful to these seminar participants, especially to Martin J. Bailey, Betty Daniel, Fuku Kimura, Kajal Lahiri, David Smyth, Avia Spivak, and Jimmy Weinblatt for useful comments and suggestions. Finally, I thank Shomu Banerjee, Robert Carpenter, Robert Chirinko, Hashem Dezhbakhsh, and Robert Utz for useful comments and discussions. All errors are mine.

Address all correspondence to: Daniel Levy, Department of Economics, Emory University, Atlanta, GA 30322-2240, U.S.A.