Chapter

Macroeconomic Dynamics and the Accumulation of Net Foreign Liabilities in the US: An Empirical Model*

Author(s):
International Monetary Fund
Published Date:
September 2005
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Introduction

The size of the external deficits run by the US through the second half of the 1990s and the beginning of the new century has raised controversial issues in the current policy debate – ranging from sustainability of US current account imbalances, to the main macroeconomic and international implications of the required adjustment in the next few years, especially as regards the rate of dollar depreciation in nominal and real terms compatible with global equilibrium. Macroeconomic and policy exercises are quite complex, as their results are sensitive to assumptions regarding basic macroeconomic parameters (e.g., expectations about growth rates and real interest rates), policy interactions among sovereign domestic policy makers (e.g., explicit or implicit currency target pursued by central banks, fiscal-monetary policy mix in different regions of the world), the structure of asset markets (including the possibility of sudden changes in the desired portfolio composition by international investors, perhaps reflecting coordination problems in financial markets) and so on. But the current policy debate has also motivated a thorough reconsideration of economic theories of the current account and external solvency, as well as of the empirical evidence on the determinants and dynamic behavior of foreign liabilities.

In this paper, we contribute to the current debate by providing novel empirical evidence on the dynamic evolution of the stock of US net foreign liabilities. Building on our previous work (Corsetti and Konstantinou [2004]), we resort to a small set of assumptions to derive an empirical model that synthesizes the joint dynamics of net debt and key macroeconomic variables such as consumption, output and investment. Using the restrictions implied by the external solvency constraint of a country, we identify empirically transitory and permanent shocks to the variables in our system, and study the dynamic response to these shocks (on the methodology, see Campbell and Mankiw [1989] and Lettau and Ludvigson [2001, 2004]).1

In the first part of the text, we will briefly summarize the results from the three variables system developed in our previous work. An important result is that our empirical model for the US appears to lend support to the basic theoretical propositions of the theory of the current account. Namely, shocks that raise output and consumption temporarily also raise the stock of net foreign assets – a pattern implied by the consumption smoothing hypothesis, but at odds with procyclical views of current account imbalances. Conversely, shocks that raise output and consumption permanently – which can be naturally interpreted as a permanent technology shock – are associated with the build up of foreign liabilities.

Our discussant during the Conference preceding this volume pointed out the possibility that the behavior of the US macroeconomy in the second half of the 1990s be affected by an asset market bubble. If this is the case, it may be possible that part of the US current account deficit has been recently financed with the sale of ultimately worthless paper to the rest of the world. This observation is relevant for our purposes insofar as our empirical results will be sensitive to the inclusion of the ‘bubble years’ in the empirical analysis. We therefore run our model on a subsample that exclude the years from 1998 on, showing that our main conclusions are substantially unaffected.

In the second part of the text, we will develop a four variables system, studying the joint dynamics of output net of government spending, consumption, investment and net foreign liabilities. Compared to the three-variable model studied in Corsetti and Konstantinou [2004], a four-variable system provides a more general empirical model of the dynamic behavior of foreign liabilities. However, the cost of generality is that it becomes harder to find an approximate expression for the capital account, which now cannot be identified separately due to the balanced growth assumptions. While the stochastic structure of the system changes, we can still identify one permanent shock (which can be interpreted as a permanent productivity shock), whereas we identify two permanent shocks in our earlier work.

Our findings in the four-variable model can be summarized as follows. First, we find strong evidence that three long-run relations exist for the variables in our system, two of which correspond to stationary ‘great ratios’ – consumption to output and investment to output ratios – while a third one is a relation between the logs of net foreign liabilities and output.

Second, by adopting Generalized Impulse Responses (GIR), we find that positive shocks that increase consumption and investment also worsen the US net foreign asset position. On the other hand, a shock that raises output leads to some improvement in the US net foreign position – although such improvement is not significant. Interestingly, we find that an exogenous shock to the stock of foreign liabilities has a marked transitory effect on the level of this variable, which accounts for a large share of its variance over short- and medium-term horizons.

Third, as in our previous work, we identify a permanent shock as an innovation with permanent effects on the variables in our system. The dynamic effects of this shock are similar to the ones studied in our three variable model: a shock with positive permanent effects on per capita output and per capita consumption also raises net foreign liabilities. In addition, consistent with the interpretation of the shock as a permanent improvement in productivity, it is associated with higher per capita investment. We find that this permanent shock dominates the variation of net foreign liabilities at horizons of six years and more, whereas transitory shocks dominate its variation at shorter horizons.

The rest of the paper is organized as follows. Section 2 reconsiders the main findings of our prior work using a different sample. Section 3 provides a further theoretical motivation for our work. Section 4 lays out our empirical methodology. Sections 5 and 6 contain the empirical results from the four variables model. The last section concludes.

The Response of US Net Foreign Wealth to Temporary and Permanent Shocks: A Review of Our Main Findings

In this section we briefly review the main features and results of the empirical methodology developed in Corsetti and Konstantinou [2004] and applied to the US, as an introduction to the extension of the same analysis presented below. The text will be developed in a non-technical way, leaving details to Corsetti and Konstantinou [2004] as well as to the next section, where a more general model is presented.

Many analysts argue that the second half of the 1990s is characterized by an asset pricing bubble driving the boom of assets markets in the US and elsewhere. A bubble could affects the intertemporal budget constraint – as the US deficit could at least in part be financed by issuing ultimately worthless assets. This consideration raises the important issue of whether our results are sensitive to the inclusion of the last part of our sample in the analysis. To address this issue, in summarizing our main results from previous work we will refer also to an application of our methodology to a sample truncated in 1997, i.e., 1963:Q1-1997:Q4. Variable definitions and a discussion of the data are provided in appendix.

Implications of Budget Constraint

In our analysis, as in Campbell and Mankiw [1989] and Bergin and Sheffrin [2000], we derive an approximate expression for the budget constraint of a country by taking a first-order Taylor approximation of the intertemporal budget constraint, imposing the appropriate transversality conditions and taking expectations. In doing so, we assume that the portfolio share of foreign wealth in domestic private wealth is stationary – so that the expected value of this share exist. Domestic private wealth is defined as the present discounted value of net output Zt, which is GDP net of government spending and investment. Let Dt denote the stock of net foreign liabilities (the negative of net foreign wealth), while Ct denotes private consumption. Referring to our previous work for detail, our procedure allows us to obtain an approximate expression for the capital account in the form:

where lower-case variables denote the log of the corresponding upper-case variables,2 and φz is a function of the expected value of the foreign wealth to domestic private wealth ratio.

The above expression defines the variable to be used in our empirical analysis. Specifically, Ct is real per-capita expenditure on nondurables and services.3 Net output, Zt, is gross domestic product net of government expenditure, investment and expenditure on durables, expressed in real, per-capita terms. The stock of net foreign liabilities, Dt, is also expressed in real, per-capita terms. We stress here that Dt records more than bonds – as it includes the whole array of assets and liabilities traded internationally. The variable Dt is derived by cumulating the current account deficits over the sample period. Data limitations do not allow us to use series of net foreign liabilities allowing for capital gains and losses on a wide array of assets – as proposed by Lane and Milesi-Ferretti [2001]. Namely, the series built in this study is at a lower frequency (annual), and for a smaller sample than the one we adopt in our work. Yet we should note here that our series and the Lane & Milesi-Ferretti series are quite correlated.4

Now, it can be shown that, under the weak maintained hypothesis that the real rate of return rt, the rate of growth of consumption and net output are covariance stationary, the budget constraint implies that the logs of consumption, net output and net foreign liabilities must be cointegrated. Even if the level of net foreign wealth is non-stationary in levels (as predicted by standard infinite-horizon intertemporal model), the transversality condition prevents it to wander away from net output and consumption. Hence, in line with the literature, KAt* is stationary.

As mentioned above, when we log-linearize the intertemporal budget constraint, we assume that φz is constant. We note here that this assumption is consistent with recent work by Kraay and Ventura [2000, 2002] and Ventura [2003], who advocate models of the current account allowing for international portfolio diversification in which the portfolio share of foreign wealth is constant. But we also note that our methodology is valid under a much weaker condition: all we need is a well-defined expected value of the ratio of net foreign debt to domestic private wealth. For instance, φz is also constant when such ratio varies over time following a stationary distribution.

Indeed, in our econometric study we are unable to reject the hypothesis that φz is constant – which may correspond to a portfolio share of foreign assets in wealth that is either time-invariant, or (more plausibly) follows some stationary distribution. With a time varying portfolio share, however, it is possible that a country switches its international net position during the sample period. Since in deriving our log-linear approximation we have assumed that no variable switches sign, a problem in applying our methodology to the US is that this country is a net creditor in the first part of our sample, and becomes a net debtor during the 1980s. Below, we will address this issue in two alternative ways. In one application of our methodology, we rescale the debt series so as to make it positive throughout the sample. In another application, we employ Dt in deviation from its sample mean. Summary statistics for the variables used are provided in Table 1.

Table 1:Summary Statistics for the Sample 1963:Q1 - 1997:Q4
ΔztΔctΔdtΔ(DtD¯D)
Univariate Summary Statistics
Mean (×102)0.57140.55351.50813.7803
Standard Deviation (×102)1.02700.47144.92796.3945
Autocorrelation0.13060.41530.87460.9447
Correlation Matrix
Δzt1.0000.4773−0.0762−0.0344
Δct.1.0000.05680.0866
Δdt..1.0000.7129
Δ(DtD¯D)...1.000
NOTES for Table 1: This table reports summary statistics for quarterly growth of net output Δzt, consumption Δct, and two measures of net foreign liabilities growth rate Δdt and Δ(DtD¯D), where all variables are expressed in real, per-capita terms. The sample spans the first quarter of 1963 to the fourth quarter of 1997.
NOTES for Table 1: This table reports summary statistics for quarterly growth of net output Δzt, consumption Δct, and two measures of net foreign liabilities growth rate Δdt and Δ(DtD¯D), where all variables are expressed in real, per-capita terms. The sample spans the first quarter of 1963 to the fourth quarter of 1997.

Our empirical approach exploits the cointegrating relation (1) – without imposing additional structure on, say, preferences of the national representative agents or technology. As long as budget constraints – which can be derived from the transversality conditions of the problem of the national representative consumers – hold, a country’s net output, consumption and net foreign debt should commove in the long-run and therefore be cointegrated. In fact, as we discuss below, our empirical findings support this hypothesis.

Specifically, we are unable to reject (marginally) the hypothesis that there is at most one cointegrating vector among net output, consumption and the stock of net liabilities (all measured in logs or, in the case of net foreign liabilities, measured in deviation from average), while we easily reject the hypothesis that there is not cointegration. The estimates of the cointegrating coefficients have the right sign and satisfy an important inequality: they imply that the value of net foreign debt is strictly smaller than the present value of net output.

The log-linearized budget constraint also implies that these coefficients should sum to minus one. A point stressed by Lettau and Ludvigson [2001, 2004] is that measurement errors may make it unlikely that these coefficients sum to minus one in empirical implementations of the model: nondurable consumption flows are not directly observable and need to be proxied; our measure of net foreign liabilities is a rough proxy of the true theoretical variable. The restriction on the cointegrating coefficients is rejected at conventional significance levels when we use our full sample 1963-2002; but it is not rejected when we consider the sample limited to the period 1963-1997. Table 2 shows the cointegrating coefficient estimates in the two applications that we consider, along with the likelihood ratio tests of the aforementioned parameter restriction.5

Table 2:Sub-Sample Long-Run Parameter Estimates and Tests
Panel A: Using Logarithms
Panel A.l: Estimated Cointegrating Relationship
β^xt=ct+β^xzt+β^ddt
ct1.166[18.324]zt+0.022[1.728]dt
Panel A.2: Restricted Cointegrating Coefficients
β^Rxt=ct+β^zzt+(β^z1)dt
ct0.998[232.116]zt0.022[0.442]dt

LR – test: Q(1) = 1.336 {0.247}
Panel B: Using dt = (DtD)/D
Panel B.1: Estimated Cointegrating Relationship
β^xt=ct+β^zzt+β^ddt
ct1.315[14.069]zt+0.014[1.414](DtD¯)/D¯
Panel B.2: Restricted Cointegrating Coefficients
β^Rxt=ct+β^zzt+(βz^1)(DtD¯)/D¯
ct0.998[434.174]zt0.002[0.609](DtD¯)/D¯

LR – test: Q(1) = 1.336 {0.247}
NOTES for Table 2: Panel A of the table reports our results using dt = log(Dt + κ) and Panel B our results using dt = [DtD)/D where D is the sample average of Dt. The numbers in square brackets are the associated t-statistics and the numbers in curly brackets are the associated p-values for the tests. The sample spans the first quarter of 1963 to the fourth quarter of 1997.
NOTES for Table 2: Panel A of the table reports our results using dt = log(Dt + κ) and Panel B our results using dt = [DtD)/D where D is the sample average of Dt. The numbers in square brackets are the associated t-statistics and the numbers in curly brackets are the associated p-values for the tests. The sample spans the first quarter of 1963 to the fourth quarter of 1997.

Dynamic Effects of Temporary and Permanent Shocks

The empirical approach in Corsetti and Konstantinou [2004] consists of using the restrictions implied by cointegration to identify the permanent and transitory components of the three-variable system. Identification is possible because cointegration places restrictions on the long-run multipliers of the shocks in a model where innovations are distinguished by their degree of persistence, as shown, for example, in Johansen [1995], King et al. [1991], and Warne [1993]. While this approach does not identify shocks that are structural in any sense, it yields results that have some natural structural interpretation.

The steps involved in the procedure are as follows. We first estimate the VEqCM, and then use the estimated parameters to back out the long-run restrictions. More specifically, cointegration restricts the matrix of long-run multipliers of shocks in the system, which identifies the permanent components. The transitory components are identified in a ‘residual’ manner. In order to study the dynamic impact of the transitory innovations, it is assumed that these are orthogonal to the permanent innovations – a description of our methodology is provided in an appendix.

In our baseline identification, the first permanent shock is the only shock that has a long-run impact on net output per capita. Hence, it has a natural interpretation as a permanent technology shock. More generally, this shock can be read as a linear combination of structural shocks that would have a permanent effect on net output. The second permanent shock in our baseline model structure has a long-run impact on consumption and foreign wealth, but no persistent effect on output. The transitory shock only affects net output in the short run, and can therefore be read as a linear combination of structural shocks that lead to transitory changes in zt – including temporary technology shocks.

Referring to our previous work for details on full sample estimation, and to Table 2 for estimates of the long-run parameters in the 1963-1997 sub-sample, we now briefly discuss two main results, regarding the response of the system to the first permanent shock and the temporary shock. The impulse responses for these shocks are shown in Figure 1 for the case of variables measured in logs, and in Figure 2 for the case in which we consider Dt in deviation from its sample mean. For the sake of comparison, Figure 1 also reports the point estimates of the impulse responses for the full sample (see Corsetti and Konstantinou [2004]).

Figure 1.Impulse Responses for sub-sample 1963-1997 (o) and the full sample (−).

Notes for Figure 1: The first permanent shock is assumed to have a long-run impact on all three variables in the system, the second permanent shock (not reported here) is assumed to have a no long-run effect on Zt, but it has a long-run impact on c¿ and dt and the transitory shock is assumed to have no long-run effect on any of the variables. The horizon in the figure is measured in years after the shock. The figure shows the response of each variable to a one-unit shock for the sub-sample 1963-1997 (o) and for the full sample 1963-2002 (–). The first column shows the responses to the first permanent shock and the second the responses to the transitory shock, all with the associated bootsrap confidence bands obtained from estimating the model for the sub-sample.

Figure 2.Impulse Responses employing (DtD) D as the Net Foreign Liabilities Variable.

Notes for Figure 2: The figure reports impulse responses for a model that includes {DtD)/D as the net foreign liabilities variable, where D is the sample average of Dt. The first permanent shock is assumed to have a long-run impact on all three variables in the system, the second permanent shock (not reported here) is assumed to have a no long-run effect on zt, but it has a long-run impact on ct and dt(DtD)/D and the transitory shock is assumed to have no long-run effect on any of the variables. The horizon in the figure is measured in years after the shock. It shows the response of each variable to a one-unit shock for the sub-sample 1963-1997. The first column shows the responses to the first permanent shock and the second the responses to the transitory shock, all with the associated bootsrap confidence bands obtained from estimating the model for the sub-sample.

First, in response to a positive transitory shock raising net output, both consumption and net output rise on impact, but consumption rises by less than net output. Correspondingly the country runs a current account surplus and accumulates foreign assets: net foreign liabilities jump down on impact and remain negative for at least ten years.

The simplest intertemporal models of the current account predicts that national agents lend abroad to smooth consumption in the face of positive temporary shocks to their net output – see e.g. Obstfeld and Rogoff [1996], chapter 2 and especially chapter 3, where in overlapping generation models the effects on foreign assets and consumption disappear over time. The pattern of our impulse response is strikingly consistent with this prediction – although our model does not allow us to reach any conclusion about optimality of the response to the shock.

We stress that our result does not lend support to procyclical current account deficits. Temporary output expansions are not associated with a widening of the external imbalance, as implied by traditional models stressing the role of real demand shocks (e.g. government spending) in generating business cycle fluctuations.

Second, in response to the first permanent shock – that, we argued above, can be interpreted as a technology shock – consumption increases on impact and keeps increasing over time, while net output increases more slowly. The permanent shock increases net foreign liabilities on impact, which keep increasing for a few years after the shock – although they revert to a lower level in the long run, but strictly above zero.

The standard intertemporal model predicts that permanent productivity shocks – that raise per capita net output and consumption in the long run – generate a current account deficit and raise net foreign liabilities: higher returns to domestic capital attract foreign investment, higher permanent income tends to reduce domestic saving. Our empirical results are once again consistent with this prediction. We will reconsider this specific result in the framework of our four variable model below.

Note that, given that the US is large in the world economy, one may expect a shock raising US productivity and US demand (i.e. reducing US net saving) to put upward pressure on the world real interest rate. To the extent that the upsurge in the US demand translates into a temporary higher real rate of return, the consumption Euler equation implies that the marginal utility of US consumption should fall gradually along the transition path. Consistent with this view, as shown by the figure the US consumption increases gradually in response to the shock.

Variance decomposition is shown in Table 3. Most of the long run variance of output and consumption is explained by the first permanent shock. Temporary shocks explain a high share of variance of output and consumption in the short run, but the influence of this shock remains relevant also over long-horizons. In addition, as in Corsetti and Konstantinou [2004], the transitory shock accounts for a respectable share to the variation in the US net foreign position. The second permanent shock explains a very limited share of the variance of net output and consumption, but its influence on the variability of net liabilities is sensitive to sample specification.

Table 3:Forecast Error variance Decomposition for the Sub-Sample
Horizon hΔzt+h – EtΔzt+hΔct+h – EtΔct+hΔdt+h – EtΔdt+h
η1tPη2tPη3tPη1tPη2tPη3tPη1tPη2tPη3tP
Panel A: Using Logarithms
10.060.100.840.490.000.510.270.540.19
20.070.120.810.540.000.440.360.500.14
30.060.140.800.580.000.420.400.490.11
40.050.160.790.620.000.380.430.480.09
80.070.200.730.730.000.260.460.490.05
120.180.200.620.800.010.190.450.510.04
160.320.180.500.850.010.140.440.530.03
200.460.150.390.880.020.100.420.560.02
400.800.060.140.930.030.040.340.650.01
Panel B: Using dt = (Dt-D)/D
10.080.180.740.470.020.510.390.370.24
20.090.210.700.510.020.470.500.340.16
30.090.240.670.550.020.430.550.330.12
40.080.260.660.590.020.390.580.330.09
80.060.330.610.700.010.290.620.330.05
120.120.360.520.780.010.210.620.340.04
160.210.350.440.830.010.160.620.350.03
200.300.340.360.860.010.130.610.360.03
400.630.210.160.940.000.060.560.420.02
NOTES for Table 3: The table reports the fraction of the variance in the h step-ahead forecast error of the variable listed at the head of each column that is attributable to innovations in the permanent shocks (P), and the transitory shock (T). Horizons are in quarters, and the underlying VEqCM is of order 1. The sample spans the first quarter of 1963 to the fourth quarter of 1997.
NOTES for Table 3: The table reports the fraction of the variance in the h step-ahead forecast error of the variable listed at the head of each column that is attributable to innovations in the permanent shocks (P), and the transitory shock (T). Horizons are in quarters, and the underlying VEqCM is of order 1. The sample spans the first quarter of 1963 to the fourth quarter of 1997.

An important result emerging from our analysis is that our main conclusions regarding the response of the system to the first permanent shock and the temporary shock are virtually unaffected by using a shorter sample, and measuring net foreign liabilities in deviations from its sample mean rather than in logs. They provide the core of what we interpret as empirical evidence that qualitatively lend support to the intertemporal approach to the current account.

A Richer Model: Long-Run Relationships Between Consumption, Output, Investment and Net Foreign Liabilities

We now develop our analysis by specifying a four variable model. Above, we have defined net output Zt as GDP net of government consumption and total investment. We now add investment as a separate variable to our system, and defined a new variable for output as GDP net of government spending – for simplicity, we will refer to it simply as output. Namely, Yt will now denote GDP net of government spending, while It will include private investment and consumption expenditure on durables (all in real percapita terms). As mentioned before, a full description of the data is provided in the appendix.

An important building block of our work is the empirical evidence on comovement, at least in low frequencies, between output, consumption and investment. This can be clearly seen from Figure 3, where the three series are plotted. Although the aforementioned variables are well characterized as integrated processes, it is usually found that the ratios of investment to output and consumption to output are usually found to be stationary (see e.g. King et al. [1991]). Essentially, there is a long tradition of empirical support for balanced growth in which output, investment and consumption all display positive trend growth but the consumption-output and investment-output “great ratios” do not.

Figure 3.Output, Consumption and Investment for the U.S.

Consistently, DSGE one-sector models restrict preferences and production possibilities so that balanced growth occurs asymptotically – implying that permanent shifts in productivity will induce long-run equiproportionate shifts in the paths of output, investment and consumption. To wit, as in King et al. [1991] suppose that total factor productivity is well described by a logarithmic random walk with drift:

Then, γA would be the average growth rate of the economy while ξt would represent deviations of actual growth from this trend. In this case realizations of ξt change the forecast of trend productivity equally at all future dates: Etlog(At+s) = Et–1log(At+s)+ξt. A positive productivity shock raises the expected long-run path and therefore there is a common stochastic trend in the logarithms of consumption, investment and output. Under these circumstances, the great ratios Ct/Yt and It/Yt become stationary. Taking logs, we have that:

which are two theoretical relations that should hold in the data.

Now, building on the framework of Corsetti and Konstantinou [2004], let us consider the intertemporal budget constraint:6

where Bt is the (initial) stock of net foreign assets, rt+j is the world real rate of return at period t+j, which may vary over time and Rt,s is the market discount factor for date s consumption, so that Rt,s=Πj=t+1s(1+rj)1 with Rt,t=1. Then by defining the stock of net foreign liabilities as Dt ≡ -Bt, the intertemporal budget constraint may be written as:

Following Campbell and Mankiw [1989], Bergin and Sheffrin [2000] and Corsetti and Konstantinou [2004], we derive an approximate expression for the above intertemporal budget constraint, by taking a first-order Taylor approximation of (6), imposing three transversality conditions and taking expectations.7 By defining rt ≈ ln(1+rt), we obtain:

where ρΨ1DtΨtΘtΨt¯,ρCΦ1exp(ctϕt¯),ρYΨ1exp(ytΨt¯)

and, ρIΘ1exp(itθt¯), while,

Our previous considerations about stationarity of portfolio shares of foreign wealth (Dt/Ψt) apply also here. By the same token, the existence of the log-linearization parameters ρΨ, ρi and ρd also implies stationarity of the present value of consumption and investment as a share of the present value of net output – corresponding to stationarity of the great ratios (see also the discussion in the appendix).

Under the realistic assumption that the rate of interest, the rate of growth of consumption, output and investment are all stationary, the left hand side of the above expression is also stationary. It should also be stressed that our approximation of the intertemporal budget constraint implies a restriction on the parameters of the right hand side of (7). More specifically, the sum of the coefficients on output, investment and net foreign liabilities should equal minus one, i.e. –(1/ρΨ)–ρiρd = –1 (see the appendix for a discussion).

But, as we explain in greater detail in the appendix, estimating a long-run relation like the left-hand side of (7) is virtually impossible, if the great ratios (3) and (4) are stationary. Furthermore, the parameters (1/ρΨ),ρi and ρd cannot be identified separately, given that (3) and (4) hold. Instead, one may aim at estimating a long-run relationship of the form:

which would essentially be a linear combination of (3), (4) and the left-hand side of (7). Thus, the parameter restriction that –(1/ρΨ)–ρiρd = –1 cannot be tested, since – as we already mentioned – the three parameters are not separately identified. In fact, if the three long-run stationary relations (3), (4) and (8) are stationary, any linear combination of the three would also be stationary. I.e., if we estimate (3), (4) and (8), taking a linear combination of them will yield a stationary relation:

Since the λ’s are arbitrary, we can chose among an infinity of linear combinations, including those satisfying the aforementioned restriction.

The empirical approach we describe below simply exploits the above cointegrating relations. We focus on the dynamics of the US net foreign position and examine, how the dynamic behavior of the net position is affected by changes in output, investment and consumption.

Data and Econometric Framework

This section describes the econometric methodology underlying our empirical approach. The problem is to analyze the dynamic behavior of a vector of variables xt with n elements. In our application, xt = (it,ct,dt,yt), where it denotes loginvestment, ct log-consumption, dt log of net foreign liabilities and yt log of output. An important point stressed in the previous section is that our main results are quite similar whether we use our full sample, or the subsample ending in 1997. Different from the analysis above, we will now carry out our study using the full sample.

Preliminary Analysis

Table 4 reports the summary statistics of the data. The standard deviation of the quarterly net foreign liabilities growth is roughly ten times as high as that of consumption growth, over four times as high as that of output growth and almost twice as high as the standard deviation of investment. Investment and consumption appear to have similar average growth rates, whereas output seems to be growing at a higher average rate. The first order autocorrelation is roughly 0.04 for investment growth, 0.42 for consumption growth, 0.28 for output growth and 0.88 for net foreign liabilities growth.

Table 4:Summary Statistics
ΔitΔctΔdtΔyt
Univariate Summary Statistics
Mean (×102)0.54660.54721.66290.6250
Standard Deviation (×102)2.91380.45344.62931.0990
Autocorrelation0.04470.41630.87850.2824
Correlation Matrix
Δit1.0000.23060.07190.8042
Δct.1.0000.05490.5179
Δdt..1.0000.0429
Δyt...1.000
NOTES for Table 4: This table reports summary statistics for quarterly growth of private investment Δit consumption Δct, private output Δyt, and the net foreign liabilities growth rate Δdt, where all variables are expressed in real, per-capita terms. The sample spans the first quarter of 1963 to the fourth quarter of 2002.
NOTES for Table 4: This table reports summary statistics for quarterly growth of private investment Δit consumption Δct, private output Δyt, and the net foreign liabilities growth rate Δdt, where all variables are expressed in real, per-capita terms. The sample spans the first quarter of 1963 to the fourth quarter of 2002.

The correlations between investment, consumption and net foreign liabilities and output growth rate are roughly 0.23, 0.07 and 0.80 respectively; the growth rate consumption is positively correlated with the growth rate of net liabilities (0.05) and output (0.52), while the growth rate of NFL is also positively correlated with output growth (0.04). Figure 4 plots the series used in the analysis, in level and growth rates.

Figure 4.The Series employed in the Analysis.

The Econometric Framework

Here we present the basic elements of our econometric modeling framework and in order to avoid clutter, we defer the reader to the appendix for a more general discussion. In our econometric analysis we use a vector autoregression (VAR) with k lags, which in it Vector Equilibrium Correction (VEqCM) form can be written as:

The key idea behind cointegration is that the matrix Π above, that pre-multiplies the levels of the variables, may not be of full rank (see Johansen [1995], Hamilton [1994]). More specifically, the hypothesis that xt is 1(1) is formulated as the reduced rank hypothesis of the matrix Π (see Johansen, [1995]), which can be decomposed into the product of two matrices αβ’, where α, β are each n×r and have full rank r<n,

Following this parameterization, there are r linearly-independent stationary relations given by the cointegrating vectors β; the matrix α gives the speed of adjustment of the endogenous variables to their ‘equilibrium’ values (the cointegrating relations), while there are also n-r linearly-independent non-stationary relations. These last relations define the common stochastic trends of the system.

For an I(0) process xt, the effects of shocks in the variables are easily seen in its Wold moving average (MA) representation,

Because uit is the forecast error in xit given past information, the elements of Cs represent the impulse responses of the components of xt with respect to the ut innovations. Since these quantities are just the 1-step ahead forecast errors, the corresponding impulse responses are sometimes referred to as forecast error impulse responses (Lütkepohl [1991]). In the case where the vector of variables xt is I(0) (i.e. stationary), Cs→0 for s→∞, hence the effect of an impulse vanishes over time and hence it is transitory.

Since the components of ut may be instantaneously correlated (i.e., the underlying shocks may not occur in isolation), orthogonal innovations are often preferred in impulse response analysis. For example, if a lower triangular Choleski decomposition is used to obtain B (Σu = BB′), the actual innovations will depend on the ordering of the variables in the vector xt so that different shocks and responses may result if the vector xt is rearranged.

For nonstationary cointegrated processes the Cs will not converge to zero as s→∞ in this case. Consequently, some shocks may have permanent effects. Distinguishing between shocks with permanent and transitory effects can also help in finding identifying restrictions for the innovations and impulse responses of a VEqCM (see Breitung et al. [2004] for an introduction to structural VEqCMs and Corsetti and Konstantinou [2004] for an application in a similar context). Specifically, if one finds r cointegrating relations and thus n-r common stochastic trends, one can identify n-r permanent and r transitory shocks employing a ‘small’ number of identifying restrictions (see King et al. [1991], Warne [1993]). Essentially, taking advantage of the assumption of orthogonality among structural shocks, one needs (n-r)((n-r)-1)/2 restrictions to identify the permanent shocks and r(r-1)/2 restrictions to identify the transitory shocks, relative to the total of n(n-1)/2 usually needed in structural VAR (SVAR) applications.

As mentioned above, the results of analyses based on orthogonalization assumptions depend on the ordering of the variables to obtain the orthogonalization. Recent work by Koop et al. [1996] and Pesaran and Shin [1998] suggested a different approach to impulse response analysis that does not require ordering of the variables. Instead of orthogonalized impulse responses from Cholesky decompositions, they propose generalized impulse responses (GIR henceforth) that are based on a “typical” shock to the system. The average response of the system to this typical shock is compared to the average baseline model where the shock is absent. Rather than examining the effect of a pure shock to, say, investment, GIR analysis considers a typical historical innovation, which embodies information on the contemporaneous correlations between the innovations.

Cointegration Analysis and Long-Run Trends

In order to analyze the dynamic behavior of our system of variables, we first need to determine the cointegrating rank. Based on univariate and multivariate misspecification statistics (reported in the appendix), we choose an empirical model with three lags. In Table 5, we report the trace test statistics for cointegration (Johansen, [1995])8 along with the asymptotic p-values (Doornik [1998]). Notice that the trace test statistics support a choice of r=3, implying the existence of one common stochastic trend (Stock and Watson, [1988]). Specifically, we reject the hypotheses that there exist n-r = 4 common stochastic trends (no cointegration), n-r = 3 common trends, n-r = 2 common trends, while we do not reject the hypothesis that there are at most three cointegrating vectors (see also the associated p-values reported in Table 5).

Table 5:Trace (Cointegration) Statistics
H0: rr = 0r ≤ 1r ≤ 2r ≤ 1
n – r4321
Q(r|n)66.7934.7215.530.22
Asymptotic p-value[0.000][0.012][0.048][0.639]
Q95(r|n)47.2129.6815.413.76
NOTES for Table 5: Q(r|n) denotes the trace statistic as defined in Johansen [1995], i.e. Q(r|n)=TΣi=r+1n In (1λ^i). The asymptotic p-values reported are calculated using the methods in Doornik [1998], while the asymptotic critical values are taken from Osterwald-Lenum [1992]. The sample spans the first quarter of 1962 to the fourth quarter of 2002.
NOTES for Table 5: Q(r|n) denotes the trace statistic as defined in Johansen [1995], i.e. Q(r|n)=TΣi=r+1n In (1λ^i). The asymptotic p-values reported are calculated using the methods in Doornik [1998], while the asymptotic critical values are taken from Osterwald-Lenum [1992]. The sample spans the first quarter of 1962 to the fourth quarter of 2002.

Having established the cointegrating rank, we need to impose some restrictions on the cointegrating vectors in order to achieve identification. The estimates of the cointegrating parameters β are obtained using maximum likelihood (Johansen, [1995]). These are summarized in Panel A of Table 6.

Table 6:Cointegrating Parameter Estimates
Panel A: Identified Cointegrating Vectors
Variableβ1β2β3
it1

(–)
0

(–)
0

(–)
ct0

(–)
1

(–)
0

(–)
dt0

(–)
0

(–)
1

(–)
yt–0.958

(0.0404)
–0.895

(0.0157)
–6.321

(0.6728)
Panel B: Overidentified Cointegrating Vectors
Variableβ1β1β1
it1

(–)
0

(–)
0

(–)
ct0

(–)
1

(–)
0

(–)
dt0

(–)
0

(–)
1

(–)
yt–1

(–)
–1

(–)
–10.708

(0.4682)
LR: Q(2) = 3.805 {0.149} (bootstrapped {0.311})
Panel C: No Long-Run Feedback Tests
H0: αj = 0ΔitΔctΔdtΔyt
LR: Q (3)

{jp – value}
23.148

{0.000}
10.138

{0.017}
19.084

{0.000}
17.359

{0.000}
WALD: W(3)

{p–value}
29.458

{0.000}
14.264

{0.003}
14.927

{0.002}
25.840

{0.000}
Panel D: Adjustment Coefficients
ΔitΔctΔdtΔyt
it – yt

[t–stat.]
–0.163

[–3.619]
0.010

[1.574]
0.118

[2.508]
–0.048

[–3.301]
ct – yt

[t–stat.]
0.142

[1.384]
0.031

[2.963]
0.465

[3.845]
0.083

[3.522]
dt – 10.708yt

[t – stat.]
–0.0003

[–0.133]
–0.0002

[–3.660]
–0.013

[–4.132]
0.002

[1.205]
NOTES for Table 6: Panel A of the table reports the Full Information Maximum Likelihood (FIML) estimates of the cointegrating vectors β subject to exactly identifying restrictions. The numbers in parentheses are the associated (conditional) standard errors. Panel B of the table reports the estimates of the cointegrating vectors subject to over-identifying restrictions. The likelihood ratio test is distributed as a Χ2 (2); the number in curly brackets are the asymptotic and bootstrapped p-value (based on 5000 replications) respectively. Panel C of the Table reports two variants of a test of no long-run feedback from the levels relations to the growth rates (also referred to as ‘weak exogeneity* test). Q (3) is the likelihood ratio test and W (3) is the Wald test with heteroscedasticity-consistent standard errors, both distributed as Χ2 (3). Finally, Panel D of the Table reports the adjustment coefficients αij along with the associated t-statistics (in square brackets). Statistically significant parameter estimates are given in boldface. The sample spans the first quarter of 1963 to the fourth quarter of 2002.
NOTES for Table 6: Panel A of the table reports the Full Information Maximum Likelihood (FIML) estimates of the cointegrating vectors β subject to exactly identifying restrictions. The numbers in parentheses are the associated (conditional) standard errors. Panel B of the table reports the estimates of the cointegrating vectors subject to over-identifying restrictions. The likelihood ratio test is distributed as a Χ2 (2); the number in curly brackets are the asymptotic and bootstrapped p-value (based on 5000 replications) respectively. Panel C of the Table reports two variants of a test of no long-run feedback from the levels relations to the growth rates (also referred to as ‘weak exogeneity* test). Q (3) is the likelihood ratio test and W (3) is the Wald test with heteroscedasticity-consistent standard errors, both distributed as Χ2 (3). Finally, Panel D of the Table reports the adjustment coefficients αij along with the associated t-statistics (in square brackets). Statistically significant parameter estimates are given in boldface. The sample spans the first quarter of 1963 to the fourth quarter of 2002.

The first two cointegrating vectors show that investment and consumption tend to commove positively and almost at a one-to-one rate with output – consistent with theoretical results from DSGE models. We also estimate a stationary relation of the form dt=6.321yt+stat. error. We can then examine whether the theoretical restrictions on the cointegrating parameters are valid, namely whether the log differences of consumption and investment with output – the ‘great ratios’ – are stationary. Panel B of Table 6 shows our results, together with the estimated cointegrating vectors. These two overidentifying restrictions imposed are not rejected at conventional significance levels. Together with the budget constraint, they imply a stationary relation between net foreign liabilities and output which now reads dt=10.708yt+stat. error. It is useful to rewrite this relationship as yt≈0.1dt+stat. error: ceteris paribus, an increase in the stock of foreign liabilities by 1 percent in equilibrium is associated with a 0.1 percent increase in the flow of output net of government spending.

Given the above estimates of the cointegrating vectors, the VEqCM representation of xt takes the form:

where Δxt, is the vector of log first differences, Δxt, = (Δit, Δct, Δdt, Δyt)’, δ is a (4x1) vector of constants, α = (αi, αc, αd, αy) is a (4x3) matrix of the adjustment coefficients and β^ is the (4x3) matrix of the cointegrating coefficients discussed above (see Panel B of Table 6).9 In general it is possible to impose some restrictions on the short-run parameters that are insignificant, and apply standard econometric systems estimation procedures such as feasible GLS.

The standard t-ratios and F-tests retain their usual asymptotic properties when applied to the short-run parameters: thus, one can impose individual zero coefficients based on the t-ratios of the parameter estimators, and eliminate sequentially those regressors with the smallest absolute values of t-ratios until all t-ratios (in absolute value) are greater than some threshold value ω. Alternatively, restrictions for individual parameters or groups of short-run parameters may be based on model selection criteria.

Using our estimated cointegrating relations (see Panel B of Table 6), we have performed a model reduction exercise starting from a model with two lagged differences of the variables. The procedure for model reduction is based on a sequential selection of variables and the AIC. Our results are summarized in Table 7.

Table 7:Parsimonious VEqCM Estimates
Panameter Estimates
[ΔitΔctΔdtΔyt]=[0.081[4.743]00.991[4.296]0]+[0.142[5.147]0.142[2.520]00.012[2.110]0.033[3.223]0.0002[4.536]0.012[2.438]0.455[3.882]0.012[4.203]0.033[4.212]0.094[4.422]0][itytctytdt10.708yt]+[0.263[3.628]1.945[4.005]00.888[4.140]0.239[2.297]0.055[1.794]000.792[19.671]0.770[4.751]00.900[5.355]00][Δit1Δct1Δdt1Δyt1]+[0.125[1.757]000.555[2.177]0.020[1.912]00.012[1.735]000.772[1.819]0000.195[1.697]0.030[2.901]0.158[2.179]][Δit2Δct2Δdt2Δyt2]Σ^u=[0.5690.0100.0480.1670.01510.0010.0150.3920.0160.083]×103,Corr(utut)=[1.000.1070.1020.7691.000.0150.4131.0000.0911.000]
System Reduction Tests
WACD: W (20) = 9.979 {0.968}, LR: Q (20) = 17.258 {0.636}
System Statistics
ΔitΔctΔdtΔyt
R20.2850.2240.7990.271
σ^it × 1002.4620.3982.0730.937
DW2.0181.9242.0131.892
NOTES for Table 7: The Table reports the estimated coefficients from a parsimonious cointegrated vector autoregressive (VAR) model, where some restrictions have been imposed on the short-run dynamics; t-statistics are given in square brackets. The Table also reports two tests for the reductions (a Wald and a Likelihood Ratio test) both distributed as a χ2 (20) and some specification statistics for each equation The sample spans the first quarter of 1963 to the fourth quarter of 2002.
NOTES for Table 7: The Table reports the estimated coefficients from a parsimonious cointegrated vector autoregressive (VAR) model, where some restrictions have been imposed on the short-run dynamics; t-statistics are given in square brackets. The Table also reports two tests for the reductions (a Wald and a Likelihood Ratio test) both distributed as a χ2 (20) and some specification statistics for each equation The sample spans the first quarter of 1963 to the fourth quarter of 2002.

First, note that formal statistical testing reveals that our reduction is valid, since both the Likelihood Ratio and Wald tests do not reject the null of valid model reduction at conventional significance levels. Second, there is quite some predictability in all the system’s equations, with adjusted R2 ranging from 0.22 (equation for Δct) to 0.79 (equation for Δdt). This is partly due to the stationary relations β′Xt−1, but also to the inclusion of lagged growth rates of all variables in our information set. The equation for net foreign liabilities displays the highest degree of predictability. Finally, all variables are shown to ‘equilibrium correct’ to at least one of the long-run relations. That is, all variables respond to the previous period’s equilibrium error β′Xt−1.

The Dynamic Behavior of Net Foreign Liabilities

In this section, we finally arrive to the core of our empirical contribution, whereas we derive and examine the dynamics of our system. Although the results presented in Table 7 already shed some light on it, an exact interpretation of the complicated dynamic behavior of the four variables is quite cumbersome. A simpler way to present the adjustment dynamics consists in presenting impulse responses and analyzing the forecast error variance decomposition of net foreign liabilities. In what follows, we will do so by exploiting different impulse-response methodologies.

We proceed in our analysis by using both the VEqCM – with no restriction imposed on the short-run parameters – and the subset VEqCM, obtained from our model reduction exercise (to which we will refer as VEqCM and subset VEqCM respectively).10 We compute impulse responses from both the full VEqCM and the subset VEqCM to investigate the dynamic effects of shocks on net foreign liabilities, providing bootstrap confidence bands for all our experiments.

Generalized Impulse Responses

Since the estimated instantaneous residual correlations are large (see Table 7), it is not appropriate to consider the forecast-error impulse responses. The first methodology we adopt is the Generalized Impulse Responses proposed in Pesaran and Shin [1998]. As mentioned above, GIRs do not require identifying restrictions – they are order free: all variables are endogenous and affect each other taking into account the historical distribution of the (reduced form) shocks. So, for example, in analyzing the effect of investment on net foreign position, GIRs enable us to study the total effect of the former variable on the latter after taking into account the implications of investment shocks for output and consumption, as well as all the feedback effects through the system. Thus our results should not be interpreted as a ‘partial effect’ of investment on net foreign liabilities, holding the rest of the variables constant. However, dynamic responses are by no means structural, in that no shock is identified in a structural sense.

The impulse responses of net foreign liabilities to a one standard deviation generalized shocks to each of the variables in the system are shown in Figure 5. Notable results are as follows. A one standard deviation rise in investment ultimately leads to a permanent increase in net foreign liabilities,11 though with some fluctuations in the short and medium run. The effect of consumption on net foreign liabilities has the same sign as investment. Note that the response appears to be significant for a year following the shock, and it becomes again significant roughly eight years after the shock. On the other hand, a one standard deviation rise in net foreign liabilities has a pronounced impact on the level of this variable, but its effects are transitory. One way to interpret this result is that there might be some inherent short-term dynamics of the US net foreign position, reflecting the behavior of international financial markets (see Figure 6). Finally, an increase in output leads to a temporary improvement of the US net foreign position (decrease in the level of net foreign liabilities).

Figure 5.Generalized Impulse Responses of Net Foreign Liabilities.

NOTES for Figure 5: The figure shows the effects of a one-standard-deviation increase in each of the variables shown in the denominator. The results are ordering independent. The horizon is in quarters after the shock. Panel A reports the results for the unrestricted VEqCM(2) and Panel B the results for the subset VEqCM (2) (see Table 7). The 95% confidence intervals were obtained by bootstrap Monte Carlo simulation (10000 replications).

Figure 6.Generalized Impulse Responses to Shocks in Net Foreign Liabilities.

Notes for Figure 6: The figure shows the effects of a one-standard-deviation increase in net foreign liabilities on each of the variables in the system. The results are ordering independent. The horizon is in quarters after the shock. Panel A reports the results for the unrestricted VEqCM(2) and Panel B the results for the subset VEqCM (2) (see Table 7). The 95% confidence intervals were obtained by bootstrap Monte Carlo simulation (10000 replications).

A similar picture emerges when examining the generalized forecast error variance decomposition of net foreign liabilities, which is reported in Panel A of Table 8. The variation of net foreign liabilities is dominated by generalized innovations to itself at all horizons. More specifically, at a horizon of one year, the generalized innovations in dt account for roughly 95% of its variation. On the other hand, at longer horizons (about 8 years) the generalized innovations in consumption seem to account for a non-negligible amount of the variation in dt while investment and output have only minor contributions to the variation of net foreign liabilities.

Table 8:Forecast Error Variance Decomposition
Panel A: Generalized Innovations
A.1: VEqCM(2)A.2: Subset VEqCM(2)
Δdt+h—EtΔdt+hΔdt+h—EtΔdt+h
Horizon hutiutcutdutyutiutcutduty
10.010.001.000.010.010.001.000.01
20.030.010.980.010.050.000.980.01
30.040.020.960.010.060.010.970.01
40.050.030.950.010.070.010.960.01
80.050.070.900.010.070.030.920.01
120.040.090.820.010.050.040.860.01
160.040.090.690.030.040.040.760.03
200.040.090.560.040.040.050.650.05
400.080.160.250.020.060.110.340.04
Panel B: Permanent and Transitory Innovations
B.1: VEqCM(2)B.2: Subset VEqCM(2)
Δdt+h—EtΔdt+hΔdt+h—EtΔdt+h
Horizon hPTPT
10.010.990.020.98
20.010.990.040.96
30.020.980.050.95
40.020.980.070.93
80.100.900.150.85
120.210.790.240.76
160.330.670.330.67
200.440.560.420.58
400.740.260.700.30
1.001.00
NOTES for Table 8: Panel A of the table reports the fraction of the variance in the h step-ahead forecast error of the Net Foreign Liabilities dt that is attributable to generalized innovations in uti, utc, utd and uty. Notice that by construction these fractions may not sum to unity. Panel B of the table reports the fraction of the variance in the h step-ahead forecast error Net Foreign Liabilities that is attributable to permanent (P) and transitory (T) innovations. Horizons are in quarters, and the underlying VEqCM is of order 2. The sample spans the first quarter of 1963 to the fourth quarter of 2002.
NOTES for Table 8: Panel A of the table reports the fraction of the variance in the h step-ahead forecast error of the Net Foreign Liabilities dt that is attributable to generalized innovations in uti, utc, utd and uty. Notice that by construction these fractions may not sum to unity. Panel B of the table reports the fraction of the variance in the h step-ahead forecast error Net Foreign Liabilities that is attributable to permanent (P) and transitory (T) innovations. Horizons are in quarters, and the underlying VEqCM is of order 2. The sample spans the first quarter of 1963 to the fourth quarter of 2002.

The Effect of Permanent (Technology) Shocks

Given that we have found three cointegrating relations in our model, we know that there is one permanent shock and three transitory shocks in the system. Following the methodology spelled out in previous work (Corsetti and Konstantinou [2004]), we now proceed by assuming that these four shocks are orthogonal. This assumption automatically identifies the permanent shock. As discussed above, since this shock raises per capita output (as well as consumption and investment) in the long run, it has a natural interpretation as a permanent technology innovation. In what follows, we focus exclusively on this shock, for two reasons. First, identifying the three transitory shocks separately in our four variables system requires further identifying assumptions – for which theory offers limited or no guidance. Second, the results of the three variables system show that permanent shocks raise net output, consumption and net liabilities. We would like to address the important question as of whether investment dynamics is consistent with our interpretation of this shock as a technology shock, i.e. whether this shock raises the rate of capital accumulation.

The effects of a one standard deviation permanent shock are shown in Figure 7. As apparent from the figure, our permanent shock raises consumption (ct), output (yt), net foreign liabilities (dt) and investment (it) rise. Strikingly, the effects of this shock are quite similar to those presented in Corsetti and Konstantinou [2004] and reviewed above – derived in the context of a quite different empirical exercise. Relative to our previous results, the valuable additional information is that investment indeed increases, consistent with our interpretation of the shock as a domestic technology shock, and in accord with standard open economy models.

Figure 7.Impulse Responses to the Permanent (Technology) Shock.

NOTES for Figure 7: The figure shows the effects of the permanent, technology shock on investment it, consumption ct, net foreign liabilities dt and output yt. The horizon is in quarters after the shock. Panel A reports the results for the unrestricted VEqCM(2) and Panel B the results for the subset VEqCM(2) (see Table 7). The 95% confidence intervals were obtained by bootstrap Monte Carlo simulation (10000 replications).

Consumption responds to the permanent shock by increasing gradually over time –possibly reflecting equilibrium changes in the real interest rates – as in our three variable model. Investment, instead, increases rapidly, reaches a peak after approximately one year, and then converges slowly to a new higher steady state level. Net liabilities also increase gradually: the change becomes significantly different from zero after one year.

Panel B of Table 8 reports the fraction of the total variance in the forecast error of Δdt that can be attributed to the permanent shock as opposed to the sum of the three transitory shocks. For a horizon between one and four quarters, the transitory shocks account for a portion between 98% and 93% of the variance in net foreign liabilities. At a horizon of eight to 20 quarters ahead, the transitory shocks continue to contribute a considerable amount to the forecast error variance of Δdt (between 85 and 58 percent). However, the permanent shock now accounts for roughly 15% – 42% of the variance.

At a horizon of forty quarters, the permanent technology shock accounts for 70% of the variance of net foreign liabilities. Notably, at a horizon of forty quarters, the transitory shocks still contribute 30% to the variance of Δdt, while the permanent shock accounts for the total of the long-run error variance in dt (and in fact for all variables).12 Again, these findings are similar to those reported in Corsetti and Konstantinou [2004], in that permanent shocks account for the majority of the variation in Δdt in the medium to long-run, whereas transitory shocks contribute a non-negligible fraction of the variation of the US net foreign position.

Conclusions

In this paper we have studied the long- and short-run dynamic behavior of the US net foreign position. The analysis was performed using quarterly data on real per-capita output, real per-capita consumption, real per-capita investment and real per-capita net foreign debt, and applying several recent developments in the econometric analysis of non-stationary processes and cointegration.

In the first part of the paper we have reviewed results of our earlier contribution (Corsetti and Konstantinou [2004]) and we have also addressed concerns about a potential influence on our result of a possible asset price bubble in the asset markets at the end of the 1990s, by considering a shorter sample. We have found that net output, consumption and the stock of net liabilities (either in logs or in deviations from the sample mean) are cointegrated, as implied by the intertemporal budget constraint. We have then identified transitory and permanent shocks to the system. As suggested by the intertemporal approach to the current account, a permanent shock that raises per-capita net output permanently (which has a natural interpretation as a permanent technology shock) also raises consumption and net foreign liabilities. Conversely, transitory improvements in net output lead to a build up of foreign asset – in contrast with traditional models predicting pro-cyclical current account balances.

In the second part of the paper, we extended our methodology to a four-variable model, including investment as a separate variable. Using a cointegrated VAR model, we uncovered three long-run relationships: two correspond to the “great ratios” discussed in recent DSGE models and are stationary, and the final one results from combining these two ratios with a log-linear version of the intertemporal budget constraint, i.e., a long-run relation between the level of US net foreign wealth and US output.

Second, we have examined the propagation of historical-typical shocks to the system by means of Generalized Impulse Responses. We found that shocks raising consumption and investment tend to worsen the US net foreign position; shocks that increase output tend to improve it mildly, whereas shocks that exogenously increase the stock of US net foreign liabilities have a pronounced but transitory effect on the level of this variable. In addition, the later type of shocks accounts for the vast majority of the fluctuations in the net foreign position at short- and medium-term horizons.

Finally, reconsidering the exercise in the first part of the paper, we have used the estimated cointegrated VAR to analyze the effect of permanent innovations in explaining the dynamics of US net foreign position. We found that a permanent technology shock raising per-capita output and consumption also increases investment and worsens the US net foreign position. Such result confirms our interpretation of the permanent shock that in our system raises long run output, as a permanent technology shock. In the four-variable model, the dynamic behavior of investment is fully consistent with the dynamics response to a permanent productivity shock that raises the returns to capital in the US. This shock accounts for the vast majority of the variation of net foreign liabilities at medium- and long-term, whereas temporary shocks account for most of its variability at shorter horizons.

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Appendix

Data Description

This appendix briefly describes the variables employed in the analysis. For all variables except net foreign liabilities, our source is the FRED II Database of the Federal Reserve Bank of Saint Louis.

• CONSUMPTION Ct

Consumption is measured as expenditure on non-durables (PCNDGC96) and services (PCESVC96). The quarterly series are seasonally adjusted at annual rates, in billions of chain-weighted 1996 dollars.

• OUTPUT NET OF GOVERNMENT SPENDING Yt

This is defined by the identity YtGDPtGt. GDPt is the real gross domestic product (GDEC1) and Gt is real government consumption expenditures & gross investment (GCEC1). All series are seasonally adjusted at annual rates, in billions of chain- weighted 1996 dollars.

• PRIVATE INVESTMENT It

Investment It is defined as real gross private domestic investment (GPDIC1) + real change in private inventories (CBIC1) + real personal consumption expenditure on durable goods (PCDGCC96). All series are seasonally adjusted at annual rates, in billions of chain-weighted 1996 dollars.

• NET OUTPUT Zt

This is ZtGDPtGtIt. GDPt, Gt and It are defined above.

• NET FOREIGN DEBT Dt

We build our series of net foreign liabilities by cumulating the negative of the US Current Account (BOPBCA). We scale the resulting series by 1.000, so that the series become positive throughout our sample. In the cumulated current account series, the minimum observation (largest negative in absolute value) is -699.77. So we have experimented using different additive constants (750, 800, 900, 1000, 1100), verifying the absence of any qualitative difference in our results.

• POPULATION

Our measure of population was obtained by sampling at the end of each quarter the monthly population series.

• PRICE DEFLATOR

To deflate net foreign liabilities, we employ the personal consumption expenditure chain-type deflator (1996=100), seasonally adjusted (PCECTPI), as a proxy of the unobserved price deflator corresponding to our measure of consumption.

Lag-Length Selection and Misspecification Statistics

Table A.1:Misspecification Statistics of VAR with k=3 lags
Panel A: Univariate Statistics
Equationσ^ϵ×100AR(12)ARCH(3)NORM(2)R2
Δit2.43370.8750.5153.7250.349
Δct0.39861.872*0.1389.608**0.275
Δdt2.01461.43217.093**72.164**0.823
Δyt0.92260.8211.5919.309**0.342
Panel B: Multivariate Tests
LM1LM4LM8LM12LB (40)NORM (6)
17.30727.98631.70211.445754.55889.684
[0.366][0.032][0.011][0.781][0.000][0.000]
Panel C: Lag-Length Selection
klog LLR(k-1/k)pvalue
0614.8208N/AN/A
11914.3062517.752**[0.000]
22057.462270.207**[0.000]
32072.62127.854*[0.033]
42084.13520.581[0.195]
NOTES for Table A.1: The R2 can be interpreted as the fit of the model for each variable relative to a random walk with drift. AR(12) is an LM test statistic for autocorrelation (F(12,136) distributed), ARCH (3) is the test for ARCH effects (F(3,142) distributed), and NORM is the Jarque-Bera test for normality (χ2(2) distributed), while * (**) denotes significance at the 5% (1%) level. The LMi are tests of i-th order autocorrelation distributed as a χ2(9). The NORM (6) is a multivariate Normality test (Doornik-Hansen, 1994) which is distributed as a χ2(6). The L-B(39) is the multivariate version of the Ljung-Box test for autocorrelation based on the estimated auto- & cross-correlations of the first [T/4=39] lags and is distributed as a χ2(333). log L denotes the value of the log-likelihood, LR is sequential (i.e. k vs k—1 lags) Likelihood Ratio test statistic corrected by a degrees of freedom adjustment. The lag order selected is given in boldface.
NOTES for Table A.1: The R2 can be interpreted as the fit of the model for each variable relative to a random walk with drift. AR(12) is an LM test statistic for autocorrelation (F(12,136) distributed), ARCH (3) is the test for ARCH effects (F(3,142) distributed), and NORM is the Jarque-Bera test for normality (χ2(2) distributed), while * (**) denotes significance at the 5% (1%) level. The LMi are tests of i-th order autocorrelation distributed as a χ2(9). The NORM (6) is a multivariate Normality test (Doornik-Hansen, 1994) which is distributed as a χ2(6). The L-B(39) is the multivariate version of the Ljung-Box test for autocorrelation based on the estimated auto- & cross-correlations of the first [T/4=39] lags and is distributed as a χ2(333). log L denotes the value of the log-likelihood, LR is sequential (i.e. k vs k—1 lags) Likelihood Ratio test statistic corrected by a degrees of freedom adjustment. The lag order selected is given in boldface.

The Log-Linearized Intertemporal Budget Constraint

The intertemporal budget constraint is given by:

where Dt is the initial (period t) net foreign debt. We can write (13) as:

where Φt = Σj=0Rt,t+jCt+j,Ψt = Σj=0Rt,t+jYt+j,andΘt = Σj=0Rt,t+jIt+j. Similarly:

Taking logs

The LHS of (15) can be approximated by taking a first-order Taylor approximation (see Campbell et al. [1997]):

Defining

we can rewrite (16) as:

and the approximate expression for (15) can be written as:

or:

Notice that:

so

Log-linearizing as above we have:

where ρCΦ=1exp(ctϕt¯) and ρ < 1. Using the trivial identity Δɸt+1 = Δct+1 + (ctɸt) – (ɸt+1ct+1) and (20), equating the LHS, we obtain a difference equation in the log(Ct/Φt) ratio. Then solving forward:

where the condition limTρCΦT(ct+Tϕt+T)0 has been imposed. Observe what the last condition implies. We have:

so to the extent that (ct+Tɸt+T) is a stationary process, the limit term will go to zero, at least in expectation (see Campbell et al. [1997]). To see this more clearly, using the definition of the log-differential:

which is a stationary process under the maintained assumption of stationary consumption growth (ΔlogCt) and stationary rates of return (rt).

Notice also, that:

and using similar steps, we obtain:

while for investment we obtain:

where ρ and ρ defined similarly to the above log-linearization parameters. Substituting (21), (22) and (23) in (19) we find that:

or by taking expectations:

where:

Notice that:

Implications for Stationary (Cointegrating) Relations

Under the assumption of balanced growth:

are stationary variables. Consider the LHS of (25) which is also supposed to be stationary:

Reparameterizing we obtain:

which is also stationary.

It is well known that linear combinations of stationary variables are stationary variables, hence subtracting from (28) equation (26) and adding ρi(ityt), we are left with:

which will also be stationary.

Notice that it is impossible to test the parameter restriction –(1/ρΨ) – ρiρd = –1, as we are only able to estimate one parameter φ, which is a function of the three parameters in our linearized model. In fact, there is an infinity of acceptable solutions that would satisfy the above restrictions and can still match the estimate of φ. To clarify this point, let us write again:

where et is stationary. With a bit of algebraic manipulation, and assuming that – (1/ρΨ) – ρiρd = –1 holds, we obtain:

If we knew ρi, provided that the long run restriction on parameters holds, then we could easily determine ρd. But without knowing ρi, we would have to solve one equation in two unknowns (hence we have an infinity of solutions).

Implications for the Coefficients ρΨ, ρi, and ρd

The coefficients ρΨ, ρi, ρd defined above all depend on Dt, Ψt and Θt. Consider for instance ρΨ. Using definitions, we can right it explicitly as:

This expression shows that ρΨ depends on the average (steady-state) share of foreign wealth Dt in domestic wealth Σj=0Rt,t+jYt+j, the mean value of the “great ratio” (It/Yt) and the relative growth rates of investment and output. Using similar steps it is also easy to see that ρi and ρd will also depend on the share of foreign liabilities in domestic wealth, as well as the “great ratio” of investment and output and their relative growth rates.

Finally, using the fact that

if the parameter ρΨ exists, with,

the means of (D/Ψ), (It/Yt) and the growth rates of investment and consumption also exist – so that

implying that the present value of investment and consumption to output will also be stationary (with mean value bounded by unity – as far as ρΨ ∊ [0,1).

Econometric Methodology

In our econometric analysis we employ a vector autoregression (VAR) with k lags, namely:

where A(L) = A1L + A2L2 +… + AkLk is a matrix polynomial in the lag operator, xt is a nxl vector of variables in the system, and δ is (n×1) a vector of constants. We assume that ut is a sequence of independent Gaussian variables with zero mean and covariance matrix Σu and we write the system as an observationally equivalent vector equilibrium correction (VEqCM henceforth) given by:

where the coefficient matrices are defined as Π=Σi=1kAiInandΓi=Σj=i+1kAj.

The notion of cointegration stems from the fact that the matrix Π above, that pre-multiplies the levels of the variables, may not be of full rank (see Johansen [1995], Hamilton [1994]). In particular, we first note that the general condition for xtI(0) (i.e. stationary) is that Π has full rank, so it is non-singular. In this case, |A(1)| = |Π| = 0 corresponding to the usual condition that all the eigenvalues of the companion matrix (or the roots of the characteristic polynomial) should lie within (outside) the unit circle (see Hamilton [1994], Lütkepohl [1991]). As stationary variables cannot grow systematically over time (that would violate the constant-mean requirement), if xtI(0), then E[xt] = μx. Taking expectations of (30) yields:

so when Π has full rank, E[xt] = –Π1δ. Thus, the levels of stationary variables have a unique equilibrium mean. When xt exhibits I(1) behavior, Π is not full rank and therefore (31) leaves some of the levels indeterminate. Conversely, when Π=0, we write the VAR in differences, Δxt - these are stationary if Γ(1)=InΣi=1k1Γi has full rank, in which case Xt ∼ I(1).

As we have already mentioned in text, the hypothesis that xt is I(1) is formulated as the reduced rank hypothesis of the matrix Π, in which case it can be decomposed into:

Following this parameterization, there are r linearly independent stationary relations and n-r linearly independent non-stationary relations, which define the common stochastic trends of the system. In this case the moving average representation (or solution) of xt as a function of the disturbances ut, the initial conditions x0, and the deterministic variables δ is given by:

where C(1) = β(α´Γ(1)β)–1α, α and β are n×(n–r) matrices orthogonal to α and β respectively, C(L) is a polynomial in the lag operator, and A is a function of initial conditions, such that β´ A = 0. It should further be noted that under the assumption of cointegration, there are n-r linear combinations of the reduced form shocks that have permanent effects on the levels variables, while there are also r linear combinations of the reduced form innovations that do not have long-run effects on any of the variables in the system. We discuss these issues below.

Under the cointegrating restrictions one can estimate a VEqCM representation for xt which takes the form:

The term β´xt–1 gives last period’s equilibrium errors; α is the vector of “adjustment” coefficients (or loadings) that tells us which of the variables react to last periods equilibrium errors (cointegrating residuals) – that is, which variable, and by how much, adjusts to restore the equilibrium relations β´xt–1 back to their mean when a deviation occurs. By virtue of the Granger Representation Theorem (GRT, Engle and Granger [1987]), if a vector of variables xt is cointegrated, then at least one of the adjustment parameters in the n×r matrix α must be non-zero in the VEqCM representation (34). Thus if xi does at least some of the adjusting needed to restore the long-run equilibrium subsequent to a shock that distorts this equilibrium, then some of the parameters in the 1×r vector αi, should be different from zero in the equation for Δxi in the VEqCM representation (34).

Dynamic Responses

For an I(0) process xt, the effects of shocks in the variables are easily seen in its Wold moving average (MA) representation,

The coefficient matrices of this representation may be obtained by recursive formulas from the coefficient matrices Aj of the levels VAR representation, A(L)xt = δ + ut (see Lütkepohl [1991] or Hamilton [1994]). The elements of the Cs’s may be interpreted as the responses to impulses hitting the system. In particular, the ij-th element of Cs represents the expected marginal response of xi,t+s to a unit change in xit holding constant all past values of the process.

Since the components of ut may be instantaneously correlated, orthogonal innovations are often preferred in impulse response analysis. Using a Cholesky decomposition of the covariance matrix E(utut´) = Σu is one way to obtain uncorrelated innovations. Let B be a lower-triangular matrix with the property that Σu=BB.. Then orthogonalized shocks are given by et = B–1 ut. Substituting in (35) and defining Di = CiB(i = 0,1,2,…) gives:

Notice that Do=B is lower triangular so that the first shock may have an instantaneous effect on all the variables, whereas the second shock can only have an instantaneous effect on x2t to xnt but not on x1t. This way a recursive Wold causal chain is obtained. The effects of the shocks et are sometimes called orthogonalized impulse responses because they are instantaneously uncorrelated (orthogonal).

A well-known drawback is that many matrices B exist which satisfy BB´ = Σu – the Choleski decomposition is to some extent arbitrary if there are no good reasons for a particular recursive structure. Clearly, if a lower triangular Choleski decomposition is used to obtain B, the actual innovations will depend on the ordering of the variables in the vector xt so that different shocks and responses may result if the vector xt is rearranged.

For nonstationary cointegrated processes the Wold representation (for the levels of the process xt) does not exist. Still the Cs impulse response matrices can be computed as for stationary processes from the levels version of a VEqCM (see Lütkepohl [1991]). Generally the Cs will not converge to zero as s → ∞ in this case and some shocks may have permanent effects. Distinguishing between shocks with permanent and transitory effects, as we discussed in text and explain below, can also help in finding identifying restrictions for the innovations and impulse responses of a VEqCM.

As we mentioned, the results of analyses based on orthogonalization assumptions depend on the ordering of the variables to obtain B and hence the orthogonalized shocks. Recent results of Koop et al. [1996] and Pesaran and Shin [1998] though have re-examined the concept of orthogonalized impulse responses, aiming to remove this shortcoming. Instead of orthogonalized impulse responses from a Choleski decomposition, they suggested generalized impulse responses (GIR henceforth) that are based on a “typical” shock to the system.

The argument about GIR may be explained as follows. Let the Vector Moving Average (VMA) representation of the n-variable cointegrated VAR model be given by:

where κ0 is a vector of constants, κ1 are the coefficients of the deterministic trend, and ut is a vector of unobserved “shocks”, where ut ∼ IIDNn(0,Σu) and let σij be a typical element of Σu. Then it holds that:

where ej is a (n × 1) selection vector with element j equal to unity and zeroes elsewhere. Then the GIR of the effect of a “unit” shock to the j-th disturbance term at time t on Δxt+h is:

and the GIR of xt+h following a shock to the j-th variable is:

where Qh=Σi=0hCi and the GIRs are measured h periods after the shock has occurred.

In general, Pesaran and Shin (1998) show that one can interpret generalized impulse responses for a stationary vector process xt as:

They also explain that in a linear system, the impulse responses will be invariant to history (the information set on which conditioning is made), and so the GIR will depend only on the composition of the shocks as defined by ζ.

As shown in Pesaran and Shin (1998), the GIR will be numerically equivalent to the standard impulse response function based on Cholesky decompositions if Σu is diagonal.

Identifying Transitory and Permanent Components: A Short Review of Methodology

It is useful to review our methodology in some detail, and explain how it is related to our application. From the GRT it follows that, under the maintained hypothesis that the growth rates in xt are covariance stationary, there exists a multivariate Wold representation of the form:

where C(L) is a n×n matrix polynomial in the lag operator. We want to map these reduced form innovations into transformed innovations et that are distinguished by whether they have permanent or transitory effects. Without loss of generality the shocks et are ordered so that the first n-r of them have permanent effects; and the last r of them have transitory effects. Following Gonzalo and Granger [1995], we define a shock etP, as permanent if limhE(xt+h)/etP0,, and a shock, etT, as transitory if limhE(xt+h)/etT=0.

Applying the methodology of King et al. [1991], as extended by Warne [1993] and discussed in Johansen [1995], the permanent and transitory innovations may be identified using the estimated parameters of the VEqCM representation of a cointegrated system. In particular, as explained in Johansen [1995], the matrix C(1) of the Wold representation (40), admits a closed-form solution in terms of the parameters of the cointegrated VAR:

Notice that the structure of this matrix is such that it maps reduced-form disturbances ut into the space spanned by the columns of α i.e. sp). The disturbances αut accumulate to the permanent component of xt, whereas transitory disturbances will be in the null-space of C(l). We can therefore define the permanent disturbances (permanent shocks) as:

Then by requiring that the permanent and transitory shocks be orthogonal to each other, we can define the transitory shocks as:

Denoting:

we have that:

Notice that in this way we have achieved both the rotation from reduced-form shocks to permanent and transitory shocks and the orthogonalization. Let D(L) = C(L)P, and et = P–1ut, the transformed Wold representation is:

Thus each element of Δxt has been decomposed into a function of n–r permanent and r transitory shocks. Essentially, taking advantage of the assumption of orthogonality among structural shocks, one needs (n-r)((n-r)-1)/2 restrictions to identify the permanent shocks and r(r-1)/2 restrictions to identify the transitory shocks, relative to the total of n(n-1)/2 usually needed in structural VAR (SVAR) applications.

Consider the three variable case with n=3 and r=1. Notice that we need no restrictions to identify the transitory shocks: the assumption of orthogonality of the shocks is sufficient. Instead, we need one extra restriction to identify the permanent shocks. Next, consider the four variable system with n=4 and r=3. In this case we need no extra restriction to identify the permanent shock, whereas we need three extra restrictions in order to identify the transitory shocks.

*Paper prepared for the special volume of the conference “Dollars, Debt and Deficits: 60 Years After the Bretton Woods”, co-organized by the IMF and the Banco de España. For useful comments we would like to thank Charles Engel, Soren Johansen, our discussant Jaume Ventura, as well as conference participants in the aforementioned Conference. This paper is part of the research network on ‘The Analysis of International Capital Markets: Understanding Europe’s Role in the Global Economy,’ funded by the European Commission under the Research Training Network Programme (Contract No. HPRN-CT-1999-00067).
Correspondence to: European University Institute, Department of Economics, Villa San Paolo, Via dellaPiazzuola 43,1-50133 Florence, Italy. Email: giancarlo.corsetti@iue.it. Tel: +39-055-4685760, Fax: +39-055-4685770.
Correspondence to: University of Rome III, Department of Economics, Via Ostiense 139, 1-00154 (RM) Rome, Italy. Email: konstant@uniroma3.it. Tel: +39-06-57374258, Fax: +39-06-57374093.
1Relative to the literature on the current account, our analysis of the net foreign position dynamics requires a minimal set of equilibrium restrictions. In the three-variable model developed in our previous contribution, the main assumption is that the external solvency constraint holds, complemented by a stationary distribution for the portfolio share of foreign wealth (see also Kraay and Ventura [2000, 2002] and Ventura [2003]). In our four-variable model, two additional conditions follow from balanced growth: the “great ratios” consumption to output and investment to output should be stationary – see King et al. [1991] among others.
2Throughout this paper we use lower case letters to denote log variables (e.g., ct ≡ ln(Ct), xt ≡ ln(Xt) etc).
3Since we are mainly interested in the net foreign liabilities dynamics generated by consumption, we prefer to exclude expenditure on durables expenditure – as they replace (or add to) capital stock rather than buying a service flow from the existing capital stock. We include durable expenditure in private investment.
4The correlation between our measure of net foreign liabilities and that reported in Lane and Milesi-Ferretti [2001] (based on adjusted cumulative current account) is 0.987. See also the discussion in Corsetti and Konstantinou [2004].
5We estimate the relation β′Xt = ctφzzt + (φz – 1)dt, where β′=[1,–φz,(φz–1)]. It is worth clarifying that there are three alternative ways of expressing this theoretical restriction: (i) the sum of all the coefficients of β should be zero; (ii) the sum of the coefficients on zt and dt should be minus one; (iii) or solving, ctφzzt + (1 – φz)dt where the sum of the two coefficients equals one. These are alternative ways of expressing the same parameter restriction.
6We have imposed the usual transversality condition limk→∞Et(Rt,t+kBt+k+1) = 0.
7The three transversality conditions are:limTρCΦT(ct+Tϕt+T)=0,limTρYΨT(yt+TΨt+T)=0,limTρIΘT(it+Tθt+T)=0,where ϕtln(Φt),Ψtln(Ψt) and θtln(Θt). Details of the derivation and discussion are provided in the appendix.
8Needles to say that similar results were obtained using the maximum-eigenvalue test statistics.
9We have examined whether the adjustment coefficients for each of the variables in the analysis, αl, are statistically significant. In Panel C of Table 6, we report two different statistics, a Likelihood Ratio and Wald statistics, which examine the null that none of the variables ‘equilibrium corrects’ in order to restore the equilibrium relations, β′Xt–l, back to their means following a shock to the system. Both types of tests indicate that all the variables show strong evidence of equilibrium correction, thereby rejecting the null. Finally, all the adjustment coefficients are reported in Panel D of Table 6.
10In the first case (VEqCM) we estimate (12) with no restrictions on the short-run parameters and deterministic terms (α, Г1, Г2, δ) as explained above. For the purpose of estimating impulse responses, results from Brüggemann, Krolzig and Lütkepohl [2002] indicate that strategies using a liberal model selection criterion (e.g. AIC) are well suited. In this method the significance of adjustment coefficients a is tested within the Johansen framework, before the VEqCM is estimated. Then further restrictions on δ, Г1 and Г2 are imposed by sequentially deleting variables with t-ratios smaller than a threshold value. Applying this method to our VEqCM model reduces the number of parameters by 20.
11We stress that, in light of our description of the GIRs, this result is not a mere reflection of a national account identity, as it takes into account simultaneously movements in all variables in the system.
12The property that only permanent shocks affect the variables in the long run, whereas transitory do not, follows from cointegration and is not specific to the rotation of the shocks we have chosen. See also Gonzalo and Ng [2001] for a discussion.

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