*Mike Konczal here. In light of the collapse of the argument for a "cliff" in debt-to-GDP ratio, the most pressing issue to figure out is what to make of any minor relationship between debt and GDP. Which way does the causation work? Arin Dube wrote about this last week. Today, Deepankar Basu, assistant professor of economics at the University of Massachusetts-Amherst, takes a deep dive into this data using time series methods. Though this will involve some complicated techniques and charts, this work is crucial for understanding the current situation. I hope you check it out!*

**Public Debt and Economic Growth in the Postwar U.S., Italian and Japanese Economies**

Deepankar Basu

A recent paper by Thomas Herndon, Michael Ash, and Robert Pollin (HAP) has effectively refuted one of the most frequently cited stats of recent years: countries with public debt above 90 percent of GDP experience sharp drop offs in economic growth. This “90 percent” result was put into circulation in 2010 by a paper written by Carmen Reinhart and Kenneth Rogoff (RR) and was heavily circulated by conservative policymakers, commentators, and economists.

I think the most important issue in the subsequent discussion in blogs and newspaper op-eds (for a quick rundown see here) is the question of causality. Does the negative correlation between public debt and economic growth rest on high levels of public debt causing low economic growth, as RR and other “austerians” claim (we borrow this term from Jim Crotty)? Or is the causation the reverse of what the austerians say, meaning low economic growth causes higher public debt? Using the HAP data set for 20 OECD countries, economist Arindrajit Dube of University of Massachusetts-Amherst has shown that (a) the negative relationship between public debt and growth is much stronger at low levels of growth, and (b) the association between past economic growth and current debt levels is much stronger than the association between current levels of debt and future economic growth. This is strong evidence for the second causation argument, where low growth leads to high debt.

While Dube has worked in a single equation framework with a panel data set, in this article, I change gears and ask a time series question instead: what useful information, if any, can one extract about the relationship between public debt and economic growth from historical data for individual countries? In particular, I ask the following question: can data on historical coevolution of public debt and economic growth in the postwar U.S., Italian and Japanese economies tell us anything useful about possible causal relationships among these two variables? To briefly summarize the results, I find that the time series pattern of the dynamic relationship between public debt and economic growth in the postwar U.S., Italian, and Japanese economies is consistent with low growth causing high debt rather than high debt causing low growth.

**Why I Chose the U.S., Italy, and Japan**

As reported in Table A-1 of the HAP paper, there are only 10 countries in the sample of advanced economies from 1946-2009 that witnessed debt-to-GDP ratios above 90. These countries generally experienced years with debt/GDP above 90 consecutively, so they form easily observable episodes. However, in the postwar period very few of these episodes exhibit notably slow growth. The U.S. from 1946-2009 has already been explained in detail here as being caused by the reduction in government spending due to demobilization from World War II.

Other than the U.S., the only two countries with debt-to-GDP above 90 percent and average growth below 2 percent are Italy and Japan, with 1 percent and 0.7 percent respectively. With the inclusion of the earlier years from 1946-1949, New Zealand’s average growth increases from RR’s reported -7.6 percent to 2.6 percent. That is why I chose to focus in this article on U.S., Italy and Japan.

For the U.S. economy, federal debt declined from its high value (more than 100 percent of GDP) in the immediate postwar years to its lowest level in the mid-1970s (less than 25 percent of GDP), thereafter increasing till the mid-1990s and falling again over the next decade or so before picking up again with the onset of the global financial and economic crisis in 2007. The growth rate of real GDP has fluctuated a lot in the postwar period, with average values being higher in the two decades after the end of WWII than after the 1980s.

The Italian economy has experienced a different pattern: low levels of public debt till the early 1970s followed by a three-decade-long increase, with contemporary debt levels remaining at historical highs. Japan witnessed a very similar pattern: low levels of public debt till the mid-1970s followed by four decades of steady increase, with contemporary levels of debt hovering at historical highs. In terms of economic growth, both Italy and Japan witnessed a gradual slowdown, even as growth fluctuated at business cycle frequencies, over the entire postwar period. Thus, for all the three countries, there is large variation over time in both the variables (public debt and economic growth), which can be exploited to investigate their dynamic interrelationships.

To motivate the analysis, in Figures 1.1, 1.2, and 1.3, I give time series plots of public debt and economic growth (year-on-year change in real GDP) for the three economies that I have chosen for this analysis: the U.S. economy between 1946 and 2012, the Italian economy between 1951 and 2009, and the Japanese economy between 1956 and 2009.

**FIGURE 1.1 (USA):** Time Series plots, for the period 1946-2012, of (a) federal debt held by public as a share of GDP (top panel), and (b) year-on-year change in real GDP (bottom panel). Source: data for debt is from Table B-78, Economic Report of the President, 2013; data for growth is from NIPA Table 1.1.1

**FIGURE 1.2 (ITALY): **Time Series plots, for the period 1946-2012, of (a) federal debt held by public as a share of GDP (top panel), and (b) year-on-year change in real GDP (bottom panel). Source: Herndorn, Ash and Pollin (2013).

**FIGURE 1.3 (JAPAN): **Time Series plots, for the period 1946-2012, of (a) federal debt held by public as a share of GDP (top panel), and (b) year-on-year change in real GDP (bottom panel). Source: Herndorn, Ash and Pollin (2013).

**Why Use a Time Series Framework**

Why do I adopt a time series framework? Adopting a time series lens allows one to use a vector autoregression (VAR) analysis, a popular time series methodology that is especially suitable for studying rich dynamic interactions among a group of time series variables. The pattern of dynamic interactions (allowing for complex lagged effects) can be nicely summarized through plots of orthogonalized impulse response functions, which trace out the effect of an unexpected change in a variable on the time paths of all the variables in the system (orthogonalizing the error makes sure that the effect of impulses to one error is not contaminated by cross correlation with other errors in the system). In other words, this allows a researcher to address the following question: how would the variables in the VAR evolve over time when impacted by an unexpected change in one of the variables, *holding other things constant*? The key phrases here are “unexpected change in one of the variable” and “holding other things constant.” How do we interpret these key phrases?

Recall that in a VAR, every variable is explained by its past values and by past values of the other variables in the system. Each equation also has an unexplained part, the random error term. Thus an impulse imparted to the error (i.e., the unexplained part) in one of the equations in the VAR, can be understood as an “unexpected change,” or change in the variable that is not explained by its own past values and past values of the other variables in the VAR. Orthogonalizing the errors, on the other hand, implies that a change in one error is uncorrelated by changes in other errors in the system. Hence, when the researcher traces out the impact of an impulse to one error, she is confident that it is not picking up effects of changes in the other errors. This is a clear advantage over cross sectional analysis of correlations among variables, where distinguishing the effects of changes in one variable from the other might be difficult.

In addition, a VAR allows each variable to be endogenous; i.e., it not only allows for lagged but also contemporaneous interaction among the variables. Thus, the researcher is not forced to take an *a priori* stand on whether a variable is exogenous (or not) as in a single equation estimation framework (where the dependent variable is, by assumption, endogenous, and some of the independent variables are exogenous).

Of course, a VAR will not, by itself, address the issue of causality; one needs to impose additional restrictions to distinguish causality from correlation (i.e., to tackle the so-called identification problem). A common identification strategy is to adopt a “causal ordering” of the variables in the VAR, which is a way to restrict some of the contemporaneous effects among the variables. If a variable is causally prior to another, this means that changes in the second variable cannot have any contemporaneous impacts on the first. In a two-variable vector autoregression (VAR), there are only two possible orderings: the first variable can be assumed to be causally prior to the second, or vice versa.

So, one can use both orderings (instead of taking a stand on which is the correct structural relationship) and see if the shape of the impulse response functions change according to the ordering adopted. If it does not, then the pattern of dynamic interaction captured by impulse response functions can be thought of as a reasonable approximation of underlying structural relationships. The point is this: if the impulse response functions display qualitatively similar shapes in both ordering of variables (and remember there are only two possibilities here), then the dynamic patterns of interaction are independent of the ordering. Either of them can be used to address the question: how does the system react to an unexpected change in one variable? This is a common empirical strategy in the time series literature, and as such we adopt it here. (This strategy becomes difficult to implement and interpret when there are more than two variables in the system, in which case theoretically motivated restrictions are imposed to get identification.)

**Two-Variable VAR Analysis for Individual Countries**

To investigate the debt-growth relationship, I estimate a two-variable VAR with an optimal number of lags (where public debt as a share of GDP and year-over-year change in real GDP are the two variables) for each of the three countries *separately*: the U.S. economy for the period 1946-2012, the Italian economy over 1951-2009, and the Japanese economy over the period 1956-2009. (I choose the “optimal” number of lags using the Akaike Information Criterion.) I find three interesting results.

**First**, the contemporaneous correlation between the errors in the two equations of the VAR is negative for each of the three countries (-0.56 for the U.S., -0.54 for Italy, and -0.30 for Japan). This suggests that unexpected changes in debt and economic growth move in the opposite direction in each of these countries. This finding is in line with existing results, both of Reinhart-Rogoff and their critics.

**Second**, I conduct Granger non-causality tests to understand lags of which of the two variables in the VAR better helps in predicting the other. Table 1 summarizes Granger non-causality test results for the three countries. The first column in Table 1 tests whether debt does not Granger-cause growth; i.e., the null hypothesis that all lags of debt enter the growth equation with zero coefficients. A high p-value indicates that the null hypothesis cannot be rejected; i.e., lags of debt do not help in predicting growth. The entries in the first column are all relatively large and show that lags of debt do not help in predicting growth with high levels of statistical significance. This is true for all three economies, and *especially for Italy* (which has a p-value of 0.81).

The second column in Table 1 tests for the opposite direction of predictability: it tests whether growth does not Granger-cause debt; i.e., the null hypothesis that all lags of growth enter the debt equation with zero coefficients. A low p-value indicates that the null hypothesis can be strongly rejected; i.e., lags of growth do help in predicting debt. The entries in column 2 are all relatively small and show that lags of growth help in strongly predicting debt for all three countries (both U.S. and Italy have p-values of 0, and Japan has a p-value of 0.04).

This finding about Granger non-causality is in line with similar results reported in 2010 by Josh Bivens and John Irons for the U.S. economy. The fact that similar results hold for Italy and Japan, which have been witnessing relatively higher levels of public debt in the past few decades, is indeed a strong rebuttal of austerian claims. It demonstrates that low growth leading to (or helping to predict) high debt is more consistent with the time series data than high debt leading to (or helping to predict) low growth. Moreover, this is true not only for the U.S. economy but also for Italy and Japan.

**Third**, I analyze plots of impulse response functions (IRF) to decipher possible directions of effects running between debt and growth for all three countries for the two possible “orderings” of the variables. Figure 2.1, 2.2, and 2.3 display the orthogonalized IRFs with the first “ordering,” where debt is assumed to be “causally prior” to growth (meaning changes in debt can have a contemporaneous impact on growth but not the other way around). Figure 3.1, 3.2, and 3.3 display the orthogonalized IRFs with the alternative ordering, where growth is assumed to be “causally prior” to debt (meaning changes in growth can have a contemporaneous impact on debt but not the other way around).

**FIGURE 2.1. (USA): **Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the U.S. economy for the period 1946- 2012 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Debt is causally prior to growth.

**FIGURE 2.2. (ITALY):** Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the Italian economy for the period 1951- 2009 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Debt is causally prior to growth.

**FIGURE 2.3. (JAPAN): **Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the Japanese economy for the period 1956- 2009 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Debt is causally prior to growth.

**Impulse Response Function: Impact of Debt on Growth**

Let us start with the first ordering. In the top panel (right) of Figure 2.1 (USA), a one standard deviation positive impulse to the debt shock (i.e., the error in the equation that predicts debt) reduces growth contemporaneously, but growth returns back to zero within a year and stays there after that. In the top (right) panel of Figure 2.2 (ITALY), a similar impulse to the debt shock reduces growth contemporaneously, and growth returns back to zero within the next two years and stays there after that (notice that the 90 percent confidence interval includes zero). In the top panel (right) of Figure 2.3 (JAPAN), a one standard deviation impulse to the debt shock reduces growth contemporaneously, but growth returns back to zero within a year and gradually falls over the next several years (though here, too, the 90 percent confidence interval includes zero).

What story do these pictures tell us? If debt has a contemporaneous effect on growth (but not the other way round), then an unexpected increase in the level of debt in any year (due, for instance, to an increase in the deficit of a government that has given a tax break) will reduce economic growth in that year, but the negative impact will be washed out relatively quickly. The system will return back to its original growth path within the next few years. The speed with which the system reverts back to its original state is quickest for the U.S, slower for Japan, and slowest for Italy.

**FIGURE 3.1. (USA):** Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the U.S. economy for the period 1946- 2012 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Growth is causally prior to debt.

**FIGURE 3.2. (ITALY):** Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the Italian economy for the period 1951- 2009 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Growth is causally prior to debt.

**FIGURE 3.3. (JAPAN):** Orthogonalized impulse response functions using a Cholesky decomposition for a 2 variable VAR (debt and growth) with optimal number of lags (chosen with AIC). The recursive VAR is estimated with annual data for the Japanese economy for the period 1956- 2009 and 90 percent bootstrapped confidence intervals are included in the IRF plots. Ordering: Growth is causally prior to debt.

Let us now turn to the second ordering. In the top panel (right) of Figure 3.1 (USA), a one standard deviation impulse to the debt shock has no contemporaneous effect on growth, but there is a positive effect on growth for the next two years. In the top panel (right) of Figure 3.2 (ITALY), a one standard deviation impulse to the debt shock has no contemporaneous effect on growth, and a fluctuating (negative and positive) impact on growth which is not very precisely estimated (the 90 percent confidence interval includes zero). In the top panel (right) of Figure 3.3 (JAPAN), a one standard deviation impulse to the debt shock has no contemporaneous effect on growth, but growth experiences a positive impact for the next three years, after which it starts falling – all of which is estimated pretty imprecisely (the 90 percent confidence interval includes zero).

How should we interpret these pictures? In this case, only Italy displays a negative impact of debt on growth; both Japan and the U.S. show mildly positive impacts of unexpected changes in debt levels (though the effects are estimated pretty imprecisely). Thus, if it were the case that the contemporaneous effect between debt and growth runs from the latter to the former (as the second ordering assumes), then increases in levels of public debt might even have a positive impact on economic growth, as witnessed in the U.S. and Japan. Why might this be the case? This might be reflecting the positive multiplier effect on output growth of a boost to aggregate demand coming from an increase in the government’s deficit. Evidence for the U.S. and Japan suggests that this effect might be non-zero, at least in the short run.

Thus, for all three countries and in both orderings, *an unexpected increase in debt in any year does not have any statistically significant negative effect on economic growth in future years*. When I allow the contemporaneous effect to run from growth to debt, the short- to medium-term impact is positive for the U.S. and Japan, though the effects are not very precisely estimated. This evidence is contrary to RR’s claim that high debt leads to low growth.

**Impulse Response Function: Impact of Growth on Debt**

Once again, let us start with the first ordering. In the bottom panel (left) of Figures 2.1 (USA), 2.2 (ITALY), and 2.3 (JAPAN), a one standard deviation impulse to the growth shock reduces debt unambiguously in the short and medium term. While debt starts returning to its initial level in the case of the U.S. economy after about five to six years, it keeps declining in the Italian and Japanese economies. (This seems to suggest that the impact of economic growth on debt levels is longer lasting in Italy and Japan than in the U.S.) The bottom panels (left) of Figures 3.1 (USA), 3.2 (ITALY), and 3.3 (JAPAN) display impulse response plots for a one standard deviation impulse to the growth shock for the second ordering. They paint a qualitatively similar picture to that seen for the first ordering.

So, what do these figures tell us? They show that an unexpected increase in economic growth (for instance, due to an increase in aggregate demand caused by expanding exports) will be associated with a decrease in levels of public debt. Hence, we can turn this picture around and infer the following: when there is an unexpected decrease in economic growth, it will be associated with an increase in the levels of public debt over the next several years. This is true for all the three countries and for both orderings of the variables in the VAR.

Moreover, unlike the effect of debt on growth (which we saw in the top panels of the figures), the effects of unexpected changes in growth on future debt levels are statistically significant (though imprecisely measured) up to about 10 years in the future. This evidence clearly supports the anti-austerian position that low growth leads to higher public debt.

**Summary**

To summarize, I find that the time series pattern of the dynamic relationship between public debt and economic growth in the postwar U.S., Italian, and Japanese economies is *consistent with low growth causing high debt rather than the high debt causing low growth*. I draw this conclusion from two types of analyses: Granger non-causality tests and an investigation of impulse response function plots.

Granger non-causality tests allow one to ask the following questions: (a) do debt levels in the past help in better predicting current economic growth, and (b) does economic growth in the past help in improving predictions of current debt levels? The evidence suggests that for the U.S., Italy, and Japan, the answer to the first question is a NO and the answer to the second is a YES.

Impulse response analysis allows one to address the following questions: (a) what is the impact of an unexpected increase in current debt levels on the future time path of economic growth, and (b) how does an unexpected decline in economic growth affect future levels of debt? The data suggests that an unexpected increase in debt levels has only a small effect on future economic growth but an unexpected decline in economic growth is associated with large and long-lasting increases in public debt levels.

Thus, empirical evidence from time series analysis of the U.S., Italian, and Japanese economies seems to bolster the critique presented by our colleagues Herndon, Ash, and Pollin, as well as Dube and others, of the Reinhart-Rogoff claim that high public debt leads to low economic growth. If anything, the evidence supports causality running in the opposite direction: low growth causes higher public debt.

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