I. INTRODUCTION

The natural rate of interest--also sometimes referred as a neutral or real equilibrium rate of interest--is commonly defined as the real short-term rate of interest consistent with stable inflation and output equal to potential. It is one of the key concepts for interpreting macroeconomic relationships and the effects of monetary policy. For example, it provides a metric for the stance of the monetary policy, which is expansionary (contractionary) when the real interest rate is below (above) the natural rate. Thus, monetary policy makers have a deep interest in estimating the level of the natural interest rate. (1)

Most of the previous work on this topic estimates the natural rate for only a single country or a specific area of the world such as the European Union. However, in light of the increasing degree of global economic integration, measuring the global natural rate is of some interest. This article tries to make a contribution in this direction. We assume that the world is fully integrated and ask the following questions: How has the world natural rate evolved over the past half century? Does it exhibit a similar pattern to the natural rate in the United States? What are the main contributors to historical fluctuations in the world natural rate? Does it tell us anything about the international interaction between the United States and the rest of the world?

In order to answer the questions above, we broadly apply a commonly used methodology first proposed by Laubach and Williams (2003) to the world, proxied by an aggregate of 20 advanced economies over the period 1961-2015. (2) Laubach and Williams use a state-space model to estimate the unobservable natural rate from observed output, inflation, and interest rate data by specifying a couple of simple macroeconomic relationships including a crucial natural rate equation relating the natural rate of interest to the trend rate of potential output growth, an investment/saving (IS) curve relating the output gap to the deviation of real interest rate from its natural level, and a Phillips curve that links the inflation rate to the deviation of output from potential.

Our specification and estimation deviate from the original Laubach and Williams model in a couple of ways. First, we omit import prices from our Phillips curve specification since we are interested in global aggregates and the world does not trade with anyone. For the same reason, the FRB/U.S. imported oil price in Laubach-Williams' Phillips curve is replaced with the world oil price proxied by the price of West Texas Intermediate crude oil. Second, we apply standard maximum likelihood methods to estimate the trend growth shock instead of the medium unbiased estimator proposed by Stock and Watson (1998). We do so because shocks to world trend growth are bigger than the respective individual country shocks, so that standard maximum likelihood methods do not suffer from the "pile-up problem" (Stock 1994) as by Laubach and Williams (2003). Third, while implementing the Kaiman filter/smoother algorithm, we set the conditional expectation and covariance matrix of the initial states with a diffuse prior instead of the generalized least squares (GLS hereafter) method proposed by Harvey (1989) because the latter tends to exacerbate the "pile-up problem."

There are several main findings to highlight. First, the world neutral interest rate has been declining for the past half century in a similar pattern as the trend growth rate of potential output. The trend potential output growth can explain over a quarter of the forecast error variance of the natural rate at all finite horizons. Nevertheless, consistent with Hamilton et al. (2016), we find that the relationship between the world natural rate and trend potential output growth is modest. The point estimate of the parameter that connects the natural interest rate with the trend growth rate is 0.458, which is less than half of Laubach and Williams' estimate of its U.S. counterpart and is not statistically significant. In addition, our estimates of the output gap pick up the Organisation for Economic Co-operation and Development (OECD) recession turning points quite accurately. The estimation of the IS curve indicates that the world natural rate gap imposes a significant contractionary pressure on the world output gap. Last, the Phillips curve indicates that the world output gap has a significantly positive effect on the global inflation, which shows that the short-run output-inflation trade-off exists at the global level.

II. MODEL SPECIFICATION

Our benchmark model broadly follows Laubach and Williams (2003). The key motivating equation in Laubach and Williams (2003) is the following version of the relationship between the real rate of interest (r) and the growth rate of consumption ([g.sub.c]) that falls out of almost any intertemporal household optimization problem:

(1) r = [sigma][g.sub.c] + [theta],

where [sigma] is the inverse of intertemporal elasticity of substitution and 0 is the pure rate of time preference. Laubach and Williams use this theoretical relationship to motivate a relationship between the unobserved natural rate of interest ([r.sup.*.sub.t]) and the annualized trend growth rate of potential output ([g.sub.t]):

(2) [r.sup.*.sub.t] = [cg.sub.t] + [z.sub.t],

where [z.sub.t] captures other determinants of the natural rate of interest such as time preference, fiscal policy, and so on.

The dynamics of the output gap are described by a backward-looking IS equation, where the output gap ([[??].sub.t]) (defined as the percentage deviation of real output [[y.sub.t]] from potential output [[y.sup.*.sub.t]]) is determined by its own lags, the lagged deviation of the real short-term interest rate ([r.sub.t]) from the equilibrium real interest rate ([r.sup.*.sub.t]) and a serially uncorrected shock ([[epsilon].sub.1t]):

(3) [mathematical expression not reproducible]

where the ex-ante real interest rate ([r.sub.t]) is constructed by subtracting the expected inflation rate ([E.sub.t] [[pi].sub.t+1]) from the nominal interest rate ([R.sub.t]).

The core consumption price inflation rate ([[pi].sub.t]) is assumed to be determined by its own lags, the lagged output gap ([[??].sub.t-1]) and the crude oil price inflation rate ([[pi].sup.O.sub.t]) (as a proxy for global supply shocks) and a serially uncorrelated shock ([[epsilon].sub.2t]):

(4) [[pi].sub.t] = [B.sub.[pi]] (L)[[pi].sub.t-1] + [b.sub.3][[??].sub.t-1] + [b.sub.4] ([[pi].sup.o.sub.t-1] - [[pi].sub.t-1]) + [[epsilon].sub.2t].

For parsimony, we restrict the coefficients on the lagged inflation terms--not rejected in our sample--to sum to one. This implies that the trade-off between output and inflation exists only in the short run. We also assume that the coefficients on the second through fourth lags are equal to each other, as are the coefficients on the fifth to eighth lags, that is, [B.sub.[pi]] (L) [[pi].sub.t-1] = [b.sub.1] [[pi].sub.t-1] + [b.sub.2] [[summation].sup.4.sub.i=2] [[pi].sub.t-i]/3 + (1 - [b.sub.1] - [b.sub.2]) [[summation].sup.8.sub.i=5] [[pi].sub.t-i]/4. This specification is similar to the Phillips curve equation in Laubach and Williams (2003), except that we omit the core import price inflation term, which they include in their specification, and replace the imported oil price with a measure of the global oil price.

Equations (3) and (4) are the measurement equations of our state space model. Turning to the transition equations, we assume that the variable [z.sub.t] representing the non-trend-growth determinants of the natural rate in Equation (2) follows a random walk:

(5) [z.sub.t] - [z.sub.t-1] + [[epsilon].sub.3t].

The potential output ([y.sup.*.sub.t]) and annualized trend growth rate of potential output ([g.sub.t]) are given by:

(6) [y.sup.*.sub.t] = [y.sup.*.sub.t-1] + 0.25[g.sub.t-1] + [[epsilon].sub.4t],

(7) [g.sub.t] = [g.sub.t-1] + [[epsilon].sub.5t].

We scale the annualized trend growth rate g, by 0.25 (as by Trehan and Wu 2007) because our output data are quarterly. We assume that the shocks [[epsilon].sub.1t] through [[epsilon].sub.5t] are serially uncorrected and uncorrected with one another. The assumption that potential output evolves according to a random walk with drift can be traced to the seminal papers by Kuttner (1992, 1994) who first proposed the use of an unobserved components approach to the modeling of potential output. Kuttner in turn motivated this assumption with a reference to Watson (1986), and, implicitly, to the enormously influential paper of Nelson and Plosser (1982) which argued that many macroeconomic time series were better characterized as difference stationary rather than trend stationary series. In a basic real business cycle model with perfect competition and flexible prices, the levels of potential and actual output will coincide. Potential output in a New Keynesian model of the sort that implicitly underlies the Laubach and Williams specification is essentially the flexible price level of output of a real business cycle model. The main difference between our specification and that of Kuttner is that we allow the long-run trend rate of growth g to vary over time also (and specifically, to evolve as a random walk without drift). The rationale for this assumption is that the long-run trend rate of growth of actual output has shown a tendency to change over time, and presumably this also reflects changes in the rate of trend. Potential output growth rates were a lot higher in Germany and Japan in the decades following the destruction of World War II than they have been since. Productivity growth (a key determinant of long-run growth) was a lot higher in most advanced economies in the quarter century up to 1973 than in the two decades after 1973. And growth in most advanced economies has been a lot slower in the years since the global financial crisis of 2007-2008 than in the years that preceded it.

As detailed in Appendix A, the model can be expressed in the form of a state-space model:

(8) [Y.sub.t] = H[S.sub.t] +A[X.sub.t] + [u.sub.t]

and

(9) [S.sub.t] = F[S.sub.t-1] + [v.sub.t],

where [mathematical expression not reproducible]

In applying the Kaiman filter to the model, standard maximum likelihood estimation of [[sigma].sub.3], the standard deviation of the shock to [z.sub.t], is biased towards zero because of the so-called "pile-up problem," which usually arises when the shock to the random walk process is of small size. (3) Accordingly, our estimation proceeds in two steps. In the first step, we use the median unbiased estimator proposed by Stock and Watson (1998) to estimate the noise to signal ratio [[lambda].sub.z] = [a.sub.3]([[sigma].sub.3]/[[sigma].sub.1]). In the second step, we impose the estimated value of [[lambda].sub.z] obtained in the previous step and estimate the remaining model parameters with standard maximum likelihood methods. (4)

The above estimation procedure deviates from the three-step method designed by Laubach and Williams (2003). Laubach and Williams (2003) find that the U.S. trend growth shock is small, so that the standard maximum likelihood estimates of the standard deviation of the trend growth shock ([[sigma].sub.5]) suffers from the "pile-up problem." Accordingly, they include an extra step to estimate the ratio [[lambda].sub.g] = [[sigma].sub.5]/ [[sigma].sub.4] with Stock and Watson's median unbiased estimation method and impose that ratio in latter steps. However, as we will show later, the world trend growth shock exhibits more volatility than the U.S. trend growth shock estimated by Laubach and Williams (2003), and is immune to the "pile-up problem." Thus, we skip the extra step and estimate the standard error of the trend growth shock together with other parameters simultaneously using the standard maximum likelihood method instead.

III. DATA

The model is estimated using quarterly data for the world from 1959Q1 to 2015Q4. Due to data availability, we proxy the world by an aggregate of 20 advanced economies: Canada, France, Germany, Italy, Japan, United Kingdom, United States, Australia, Austria, Belgium, Finland, Greece, Ireland, Netherlands, Norway, Portugal, South Korea, Spain, Sweden, and Switzerland. These 20 countries account for a substantial fraction of global economic activity. Moreover, the set of countries are all market economies and exhibit a high degree of economic and financial integration with each other, which justifies the assumption underlying our model. The practice of proxying the world with an aggregate of a group of mainly industrialized economies has a long precedent in the open economy macroeconomics literature. A good recent example of this is the paper by King and Low (2014) where they present estimates of what they call the "world" real interest rate based on data from government bonds that are issued with inflation protection by the G7 countries. (In fact, their preferred estimates use data from only six of the countries on the grounds that the real rate for Italy may have been unduly influenced by issues related to the potential for default or exit from European Economic and Monetary Union.) Ciccarelli and Mojon (2010) present estimates of what they term "global" inflation based on data for 22 mainly advanced OECD countries. D'Agostino and Surico (2009) ask whether "global" liquidity can help forecast U.S. inflation, and then define "global" liquidity in terms of growth rates of measures of broad money for the G7 economies. There are many other examples.

Nevertheless, we recognize that major emerging market economies, especially the BRICS (Brazil, Russia, India, China and [sometimes] South Africa), have played an increasingly important role in the global economy in recent decades. But including those countries into the analysis is challenging for a variety of reasons. The first has to do with history. Our sample period runs from 1959 through 2015. For more than half of that period, countries like China and Russia simply did not participate in the global economy in any meaningful way. Things began to change gradually in China in the late 1970s and in Russia in the early 1990s. Second, to the extent that these countries did become integrated into the global economy in the 1990s, it was primarily through trade rather than financial linkages. Financial globalization (as proxied by growth in Lane and Milesi-Ferretti's 2007 estimates of foreign assets and foreign liabilities as a share of gross domestic product [GDP]) has proceeded at a much faster pace among the industrial countries than it has among the developing countries. Many developing and emerging market economies still have extensive capital controls in place. For example, the recent paper by Rebucci et al. (2015) classifies the degree of openness/closedness of the capital account of different countries as being either "open," having controls that amount to a "gate" or having controls that amount to a "wall." Brazil and Russia fall in the gate category, while China and India are classified as having controls that amount to walls. According to the classification presented in their paper, no major developing or emerging market economy falls in the "open" category. Third, for many developing or emerging market economies, consistent time series data only become available in the early 1990s. And even then, many of these countries tended to experience extreme events. For example, Brazil experienced a hyperinflation in the early 1990s and another episode of very high inflation in the early 2000s. Mexico also experienced episodes of very high inflation in the early 1980s, in the late 1980s, and then again in the mid-1990s (in the aftermath of the Tequila Crisis.) Annual inflation in Turkey did not fall below 25% between the 1980s and the early part of this century, and peaked at rates in excess of 125% in the mid-1990s, and so on. However, in our robustness discussion below we explore the sensitivity of our estimates of the world real rate to the inclusion of such data for the BRICS as is available.

The aggregated GDP data for our baseline estimation are constructed by adding up the GDP series (measured in constant purchasing power parity [PPP] dollars) of each individual country. The nominal world interest rate and inflation rate are derived by taking weighted averages of the corresponding indicators for individual countries with the time-varying PPP-adjusted GDP shares displayed in Figure 1 as the weights. (5) The GDP shares are calculated by the ratios of the real GDP of the individual countries to the aggregated GDP of the 20 countries included in our sample. We compute the expectation of world inflation rate four quarters ahead from a univariate autoregression (AR) (Berger and Kempa 2014) model of inflation estimated over the 80 quarters prior to the date at which expectations are being formed. (6) Then, we construct the ex-ante real interest rate by subtracting the world expected inflation from the nominal world interest rate. We use the West Texas Intermediate oil price as a measure of the global oil price. The sources and construction of the data are detailed in Appendix B.

IV. RESULTS

The second column of Table 1 reports the estimates of parameters. While our data start in 1959Q1, our estimates start in 1961Q1 because of the lags in the model. To facilitate comparison with previous studies of U.S. natural rate, we also update the estimation of the model in Laubach and Williams (2003) to 2015Q4, with the parameter estimates listed in the third column of Table 1. (7)

Similar to other individual country studies, we find the world output gap to be a fairly persistent process. The summation of the autoregressive parameters in the IS equation, [a.sub.1] and [a.sub.2], is as high as 0.922. The coefficient relating the output gap to the real rate gap ([a.sub.3]) is negative and statistically significant, which indicates that a positive world real interest rate gap is indeed contractionary.

In terms of the evidence on inflation, we find that the slope of the Phillips curve ([b.sub.3]) is significantly positive as is predicted by standard economic theory. Our estimated value of [b.sub.3] (0.159) is four times the size of its U.S. counterpart (0.040). (8) One possible reason for this is that Phillips curve equations estimated using individual country data insufficiently capture the role that foreign slack plays as a driver of domestic inflation dynamics, especially in a more open economy. Borio and Filardo (2007) argued that the well-documented decline in the sensitivity of inflation to measures of domestic slack in recent decades was due in no small part to the rise of globalization, and showed that proxies for global economic slack enhanced the explanatory power of traditional Phillips curve regressions. (9) As the world becomes more integrated through trade and factor flows, domestic slack should matter less for domestic inflation dynamics and global slack should matter more. Another way of thinking about this is in the context of an individual country: inflation in Texas, for example, depends less on the level of resource utilization or unemployment in Texas and more on the level of resource utilization or unemployment in the United States as a whole, seeing as firms in Texas can access workers and resources across the United States. The reduction of barriers to international trade and factor movements (primarily capital) as a result of globalization means that the same should be true at the international level as well. Martinez-Garcia and Wynne (2010) formalized this idea in the context of a standard New Keynesian model, referring to it as the "global slack hypothesis." A direct corollary of this hypothesis is that the relationship between slack or the output gap and inflation should be stronger at the global level than at the domestic level (since the world as a whole is a closed economy), which is what our parameter estimates suggest.

Lastly, for the natural rate equation, the link between the world natural rate and the world trend growth is weak. The point estimate of the parameter c is 0.458, which is only one third of its U.S. counterpart. By contrast to the U.S. estimate, the parameter c is insignificantly different from zero, which indicates that the relationship between the natural rate and the trend growth rate is modest. This finding is consistent with the findings of Hamilton et al. (2016), who draw a similar conclusion by studying the simple cross-country correlation between the average GDP growth rate and the average real interest rate.

For the estimates of the standard errors, the shock to trend growth rate ([[epsilon].sub.5]) is more volatile than its U.S. equivalent. The standard deviation of the trend growth shock ([[sigma].sub.5]) equals 0.171, which is more than four times its U.S. counterpart. The large size of trend growth shock makes it possible to avoid the "pile-up problem" in estimating [[sigma].sub.5] which usually arises in single country studies. On the other hand, the standard deviation of the other determinants of the natural rate ([[sigma].sub.3]) is 0.127 which is around one half of the U.S. estimate. The estimates of the standard errors shed some light on the driving forces underlying the natural rate. By combining Equations (2), (5), and (7), the natural rate of interest ([r.sup.*.sub.t]) follows a random walk:

(10) [r.sup.*.sub.t] = [r.sup.*.sub.t-1] + [".sub.rt],

where the shock to the natural rate equation [[epsilon].sub.rt] = c[[epsilon].sub.5t] + [[epsilon].sub.3t]. Given the estimates above, the standard deviation of the world natural rate shock [[sigma].sub.r] = [square root of [c.sup.2] [[sigma].sup.2.sub.5] + [[sigma].sup.2.sub.3]] is 0.149 while the standard deviation of the corresponding U.S. shock is 0.254. Thus, our estimation suggests that the shock to the world natural rate is of smaller size than the shock that drives the U.S. natural rate. Furthermore, the forecast error variance of the natural rate contributed by the trend growth at all finite horizons, measured by [c.sup.2] [[sigma].sup.2.sub.5]/ [[sigma].sup.2.sub.r], is 27.6%, which is much greater than the respective U.S. ratio of 4.8%. Thus, a substantial amount of the variation in the world natural rate is contributed by the world trend potential output growth.

Figure 2 plots our two-sided estimates of the world output gap, where the shaded areas indicate recession periods as defined by the OECD. (10) It turns out that the estimated output gap picks up the business cycle turning points quite accurately. The output gap decreases significantly in each of the OECD recessions. In particular, the world output gap decreases most sharply during the global oil crisis of 1973M5-1975M5 and 1979M9-1982M12 as well as the recent 2007M12-2009M5 global financial crisis.

Figure 3 displays our two-sided estimates of the growth rate of potential output in blue dashed lines along with trend growth in black solid lines. The world potential output growth rate fluctuates around the trend growth rate as expected. It becomes less volatile between the mid-1980s and 2007, which corresponds to the so-called Great Moderation period in the United States. The potential output growth rate reaches its trough at a historically low value of -1.2 in 2009Q1 during the global financial crisis, which was the worst recession since World War II. The world trend growth rate captures the low-frequency movement in the potential output growth, which has been declining since the mid-1960s until the recent global financial crisis. The annualized trend growth rate drops from 4.8% in 1966Q1 to 0.8% in 2009Q1 and then recovers slowly to 1.2% in 2015Q4 at the end of our sample. Based on the discussion above, our estimates of the output gap, potential output, and trend growth are consistent with global economic history, which provides some support to our estimates of the natural rate.

Figure 4 depicts the two-sided estimates of the natural rate of interest in black solid lines together with the historical realization of the exante real interest rate in blue dashed lines. Similar to the single country estimates of Laubach and Williams (2003), the world natural rate of interest has been trending down during the past half century. The declining pattern is also consistent with the world real yields on 10-year government bonds estimated by King and Low (2014), which is plotted in red in Figure 4. Based on our estimates, the real interest rate lies below the natural rate for most of the period prior to 1980, which has expansionary effects on output. This loose monetary policy helped raise the global inflation rate in the 1970s as documented in Ciccarelli and Mojon (2010). In the late 1970s, the central banks in major advanced economies raised policy rates to fight inflation. The real interest rate exceeded the natural rate starting in 1980Q2 and the real interest rate gap reached almost 4.6% points in 1982Q3. This positive real interest rate gap, signifying the contractionary stance of world monetary policy, persisted until 2001Q4. The natural rate started to decline more significantly in the late 1990s and kept falling even as the real interest rate rose from 2004Q2 to 2007Q3. The divergent movement in the real interest rate and the natural rate created a big 2.1% point real interest rate gap in 2007Q3, which was followed by the global financial crisis and the Great Recession. The natural rate drops to a historically low level of 0.2% in 2009Q3 and then recovers slowly until the end of our sample in 2015Q4. During and after the Great Recession, major central banks lowered their policy rates and launched quantitative easing (QE) programs to support economic activity. In light of the low natural rate, global monetary policy was not overly aggressive but necessary to help the world economy recover from the Great Recession.

The natural rate equation above shows that the world natural rate is determined by two factors: the world trend growth rate [g.sub.t] and the other determinant [z.sub.t]. Figure 5 displays the natural rate along with the contribution of each of the underlying determinants. Most of the fluctuation in the world natural rate is determined by the trend growth rate while the other determinant ([z.sub.t]) plays a rather limited role. This is quite different from the previous estimates of the U.S. natural rate, where the other determinant ([z.sub.t]) acts as a significant contributor to the natural rate, especially in recent years as is shown in Figure 6. A possible explanation to account for such a difference is that much of the [z.sub.t] for the U.S. natural rate is contributed by the trend growth of the rest of the world. Nevertheless, to further verify this possibility requires a two-country model where the natural rate in the United States is determined by both home country trend growth and the foreign country trend growth as explored in Wynne and Zhang (forthcoming), which is beyond the discussion of this article.

V. ROBUSTNESS ANALYSIS

In this section we consider the robustness of our results to a number of deviations from our baseline specification. First, since our estimated value of the c parameter differs from what Laubach and Williams estimated for the United States, we explore the implications for our estimates of the world natural rate of simply setting the c parameter equal to the value estimated by Laubach and Williams. We also explore the implications of setting the value equal to 1, which would correspond to an assumption of log utility, and follow what Holston, Laubach, and Williams (2017) do in their international comparison of natural rate estimates. Second, we explore the implications of estimating the model in per capita terms given the significant variation in population growth rates that occurred over our sample period. And third, we explore the implications of including such limited data as there are for the BRICS countries.

A. Sensitivity to Different Values of c

Table 2 reports the estimated parameters of the model when we impose c - 1 or c = 1.321 which is the value we estimate using the Laubach and Williams methodology on just U.S. data. Note that the imposition of these different values for c makes very little difference to the estimated value of the other parameters of the model: some are the same to the second or third decimal place. However an exception is [[sigma].sub.3] (the standard deviation of the shock to the other [nontrend growth] determinants of the natural rate), which increases from 0.127 to 0.269. Our estimates of the trend growth rate [g.sub.t] are robust to different values of c. Figure 7 shows the consequences of imposing the different values for the c parameter for our estimates of the world natural rate. Perhaps not surprisingly given the results in Figure 5, the higher imposed values for c generate much higher estimates of the world natural rate in the earlier part of our sample (when trend growth rates were much higher) than in the later part of our sample (when trend growth rates were lower). Comparing the actual real rate of interest with the natural rate estimated assuming c = 1.321 implies an extraordinarily accommodative stance of global monetary policy from the early 1960s through the early 1980s. From about the early 1990s on, the three sets of estimates track each other closely, at least up to the onset of the global financial crisis, when they diverge and the estimates with the higher assumed values for the c parameter become negative, as do measured real rates. The key feature of the data that leads the estimated model to prefer a lower value of c appears to be that while the trends in output growth in the United States and the world were quite similar over the sample period, real interest rates exhibited quite different patterns. Specifically, our measures of real interest rates are a lot higher for the United States during the 1960s and the 1970s than they are for the world, whereas they are more similar in the latter half of the sample.

B. Demographics

In our baseline model, we have shown that the natural interest rate and trend rate of growth of output have been declining over our sample period. One assumption implicitly underlying the baseline model is that the population growth is stable across the sample period. However, as shown in Figure 8, world population growth declined significantly from an annual rate of 1.2% in 1961 to 0.4% in 2015. In order to examine whether this considerable shift in demographic factors contributed to the decline in the trend growth and thus the natural rate, we implement a robustness check where we redefine [y.sub.t] in the baseline model as the output per capita. (11)

The third column of Table 3 displays the parameters estimated with the output per capita data. Most of the parameters are very close to the baseline case. The exception is the parameter c that connects the natural rate with the trend growth is now 0.6, which is 31% bigger than the baseline estimate but still insignificantly different from zero. As a consequence, trend potential output growth contributes more to the volatility in the natural rate of interest than in the baseline model. The unconditional forecast error variance of the natural rate attributable to trend growth, measured by [c.sup.2][[sigma].sup.2.sub.5]/ [[sigma].sup.2], rises from 27.6% to 42.8%.

The output gap estimated using the per capita data matches very closely to the output gap estimated by the baseline model. Moreover, the trend growth rate of the potential output per capita is lower than the baseline potential output trend growth rate as expected where the gap diminishes in recent years as the population grows more slowly. Finally, Figure 9 shows that the per capita estimates of the natural rate tracks the baseline estimates very closely for most of history. The two estimates diverge most substantially in 2009Q1 when the natural rate in per capita model reaches its trough at 0% compared to 0.2% in the baseline model. Nevertheless, the baseline estimates of the world natural rate are by and large robust to the historical demographic shifts.

C. Inclusion of BRICS

A potentially more significant omission from our analysis is the failure to include data for the increasingly important BRICS countries in the global aggregates. As noted above, many of these countries that account for an increasingly important share of global economic activity played little or no role in the global economy during the 1960s, 1970s, and 1980s when the Cold War was at its height. As the Cold War ended and more countries adopted outward looking policies, these formerly excluded countries came to play an increasingly important role in the global economy. However, the transitions were not always smooth: many of these countries experienced economic crises that limit the usefulness of their economic data for inclusion in global aggregates. However, by the late 1990s most of these countries had successfully integrated into the global trading system and were playing a greater role in global economic developments.

To see how robust our findings are to the broadening of our definition of the world to include some of the larger emerging market economies, we re-did our estimation with a sample of 25 countries that includes in addition to the 20 advanced economies included in our baseline estimate the five BRICS countries (Brazil, Russia, India, China, and South Africa). This group of countries accounted for about 17% of global output in 1992, according to International Monetary Fund (IMF) estimates; by 2015, that share had increased to just under 31%. (12) Column 4 of Table 3 reports the estimated model parameters when we include the BRICS in our global aggregate. Note that the sample period is much shorter than for our baseline model, and only runs from 1999Q1 to 2015Q4. (13) Many of the parameter estimates are qualitatively similar to our baseline estimates; the most noteworthy change is to the estimated value of c, which increases from 0.458 using just the data for the advanced economies to 1.466 using the extended set of countries. The alternative estimate also happens to be closer to the number estimated for the United States (shown in Table 1).

Figure 10 shows our measure of the world real interest rate for the broader grouping of countries along with our estimate of the world natural rate. Note that the measured world real rate does not go negative in the post financial crisis period, unlike the rate shown in Figure 4. Rather it hovers around zero, which is not too surprising as it is an average of the mainly negative real rates in the advanced economies and the higher positive real rates in the BRICS. The estimated world natural rate becomes mildly negative in 2008 but quickly returns to positive territory and by the end of 2015 was close to 2%. Contrast that with the estimates shown in Figure 4 for the aggregate based on just the advanced economies, which was less than 0.5% by the end of 2015.

An obvious question is how much of the difference in the estimates of the world natural rate shown in Figure 10 is due to the inclusion of the BRICS countries in the global aggregate, and how much is due to the estimation of the model using a much shorter sample of data. Figure 11 attempts to shed some light on this question by showing four different estimates of the world natural rate over the 1999-2015 period: our baseline estimates from Figure 4 which are based on an aggregate of just advanced economies (labeled "AE" in Figure 11), our estimates based on the aggregate that includes the BRICS countries (labeled "AE + BRICS"), an estimate for the advanced economies aggregate using the parameter estimates for the full 1961-2015 sample (labeled "AE-II"), and an estimate for the advanced economies aggregate based on coefficient estimates for the 1999-2015 period (labeled "AE-III"). This shows that using the shorter sample of data makes a significant difference to the estimates of the natural rate, which should not be too surprising. The period from 1999 to 2015 was different in many ways from the period that preceded it, incorporating as it does the global financial crisis and the extended period of ultra-low interest rates that followed it. Thus while we attribute some of the difference between our baseline estimates and the estimates based on the aggregates including the BRICS to the inclusion of the latter group of countries, we also attribute some of the difference to the shorter sample period.

VI. CONCLUSION

A growing literature utilizes unobserved components models to estimate the equilibrium rate of interest by means of multivariate trend-cycle decompositions. However, most such models focus on either an individual country or a specific area like the European Union. In this article, we contribute to the literature by jointly estimating the world natural interest rate, potential output, and the trend growth rate of potential output using an unobserved components model broadly following Laubach and Williams (2003). We find that both the world natural interest rate and the trend potential output growth rate have been declining significantly in the past 50 years. The global trend growth rate of potential output contributes substantially to the variation in the global natural real interest rate. Nevertheless, our estimation shows that the relationship between the world natural rate and the world trend growth rate is modest. The estimates of the natural interest rate are robust even while controlling for demographic shifts.

By comparing the determinants of the natural rate in the United States and the world, we find that the other determinants of the natural rate in Laubach and Williams (2003) might be mostly contributed by the trend growth in the rest of the world. However, formally testing this inference requires a two-country model (or more likely a multi-country model), which is beyond the discussion of this article and is left for future research.

The biggest challenge for future research is properly incorporating the rapidly growing emerging market economies into the analysis. In our robustness discussion, we reported the results we obtain when we include the BRICS countries in our global aggregate and re-estimate our model over the period 1999-2015. The shorter sample period is dictated by data availability for these countries and the need to avoid episodes of extremely high inflation. While the results we obtain are plausible, it is hard to disentangle the effects on our estimates of broadening the aggregate to include the BRICS countries from the consequences of working with a much shorter sample of data. Our guess is that the most fruitful way forward may be to use an unbalanced panel framework with time varying coefficients.

APPENDIX A. THE STATE-SPACE REPRESENTATION OF THE MODEL

Space form:

(Al) [Y.sub.t] = H[S.sub.t] + A[X.sub.t] + [u.sub.t]

(A2) [S.sub.t] = F[S.sub.t-1] + [v.sub.t].

Here, [Y.sub.t] and [X.sub.t] are respectively vectors of contemporaneous endogenous, and of exogenous and predetermined variables. [S.sub.t] is the vector of unobserved states. The vectors of stochastic disturbance [u.sub.t] and [v.sub.t] are assumed to be Gaussian and mutually uncorrected with mean zero and covariance matrices R and Q, respectively.

The vector of observables [Y.sub.t] is given by:

(A3) [Y.sub.t] = ([y.sub.t], [[pi].sub.t])',

where [y.sub.t] denotes 100 x log real GDP and [[pi].sub.t] denotes inflation. The predetermined and exogenous variables are:

(A4) [X.sub.t] = ([y.sub.t], [y.sub.t-1], [y.sub.t-2], [r.sub.t-1], [r.sub.t-2], [[pi].sub.t-1], [[pi].sub.t-2,4][[pi].sub.t-5,8][[pi].sup.0.sub.t-l - [[pi].sub.t-1])' .

where [r.sub.t] is the real interest rate, [[pi].sub.t-j,k] is shorthand for the moving average of inflation between dates t - k and t -j and [[pi].sup.0.sub.t] is oil price inflation. The state vector is:

(A5) [S.sub.t] = [y.sup.*.sub.t] - [y.sup.*.sub.t-1], [y.sup.*.sub.t-2] [g.sub.t-1], [g.sub.t-2], [z.sub.t-1], [z.sub.t-2])',

where [y.sup.*.sub.t] is 100 xlog potential GDP, [g.sub.t] denotes the trend growth, and [z.sub.t] represents other determinants of the natural rate. The coefficient matrices are:

(A6) [mathematical expression not reproducible]

(A7) [mathematical expression not reproducible]

(A8) [mathematical expression not reproducible]

(A9) [mathematical expression not reproducible]

(A10) [mathematical expression not reproducible]

The signal-to-noise ratio [[lambda].sub.z] is estimated with the median unbiased method introduced in Stock and Watson (1998). Given [[lambda].sub.z], the vector of parameters to be estimated by maximum likelihood is [THETA] = ([a.sub.1], [a.sub.2], [a.sub.3], [b.sub.1], [b.sub.2],[b.sub.3],[b.sub.4], c, [[sigma].sub.1], [[sigma].sub.2], [[sigma].sub.4] x [[sigma].sub.5]).

APPENDIX B. DATA SOURCES

This appendix describes the data used in this project. The data are constructed by aggregating quarterly data from 1959Q1 to 2015Q4 for 20 advanced countries: Canada, France, Germany, Italy, Japan, United Kingdom, United States, Australia, Austria, Belgium, Finland, Greece, Ireland, Netherlands, Norway, Portugal, South Korea, Spain, Sweden, and Switzerland.

The variable y refers to the log of aggregated PPP-adjusted real GDP (seasonally adjusted at annual rate) measured in millions of 2011 U.S. dollars. The aggregated data are obtained by taking the sum of the real GDP from each of the individual countries. Except for South Korea, the PPP-adjusted real GDP data are available from the OECD Quarterly National Accounts dataset (OECDNAQ) in Haver Analytics. For South Korea, the PPP-adjusted real GDP data from OECDNAQ only goes back to 1970. Nevertheless, the real GDP data in local currency from 1960 to 1970 are available from the Emerging Market dataset (EMERGEPR). We combine the two series by adjusting the observations of earlier periods with the formula: [y.sup.EMERGEPR.sub.t] ([y.sup.OECDNAQ.sub.1970Q1]/[y.sup.EMERGEPR.sub.1970Q1]) for f from 1960Q1 to 1969Q4.

The aggregated nominal interest rate is the weighted average of the quarterly average annualized short-term interest rate in each individual country using GDP share as the weight. (14) We use the central bank policy rate for most of the countries. (15) For the rest of the countries, we use money-market rates instead due to the lack of availability of the central bank policy rate. For the Eurozone countries, we splice their old interest rates with the Main Refinancing Rate in 1999Q1 when the European Central Bank was formed. The only exception is Greece which joined the Eurozone in 2001 so that we stack the earlier Bank of Greece Bank Rate with the European Main Refinancing Rate in 2001Q1.

The aggregated core inflation rate is created by taking a weighted average of the annualized quarterly growth rate of each country's seasonally adjusted core consumer price index (CPI) using the GDP share as weights. For many countries, the core CPI is unavailable back to the 1960s. Statistical agencies only began to develop measures of core inflation in response to the commodity price shocks of the 1970s. As a result, we proxy the core CPI inflation rates with the CPI inflation rate when the former rates are missing.

To construct the ex-ante real interest rate, we compute the expectation of average aggregate inflation over the four quarters ahead from a univariate AR (Berger and Kempa 2014) of inflation estimated over the 80 quarters prior to the date at which expectations are being formed. In practice, because of the limited sample, for the first 20 years we use the data from 1959 to 1981 to estimate the coefficients of the AR model. After 1981, the AR model is estimated using a rolling window with the size fixed at 80 quarters. Finally, the oil price is the West Texas Intermediate spot oil price (HAVER mnemonic PZTEXP@USECON).

All the data, except for the early CPI of Ireland, (16) are from Haver Analytics. To facilitate replication of our results, we list the Haver mnemonics in the following:

Data for Baseline Advanced Economy Estimates

Real GDP. Canada: B156GDPC@OECDNAQ; France: B132GDPC@OECDNAQ; Germany: B134DPC@ OECDNAQ; Italy: B136GDPC@OECDNAQ; Japan: B15 8GDPC@OECDNAQ; United Kingdom: B112GDPC@OE CDNAQ; United States: BI 11GDPC@0ECDNAQ; Australia: B193GDPC @ OECDNAQ; Austria: B122GDPC @ OECDNAQ; Belgium: B124GDPC@OECDNAQ; Finland: B172GDPC@OECDNAQ; Greece: B174GDPC@ OECDNAQ; Ireland: B178GDPC@OECDNAQ; Netherlands: B13 8GDPC @ OECDNAQ; Norway: B142GDPC @OECDNAQ; Portugal: B182GDPC@OECDNAQ; South Korea: S542NGPC @EMERGEPR(prior 1970Q1), B542GDPC@OECDNAQ(post 1970Q1); Spain: B184GDP C@OECDNAQ; Sweden: B144GDPC@ OECDNAQ; Switzerland: B146GDPC@OECDNAQ.

Interest Rate. Canada: Central Bank Rate, C156FROS @OECDMEI; France: Overnight Interbank Rate, C132FRUO@ OECDMEI; Germany: Overnight Interbank Rate C134IM@IFS; Italy: Discount Rate, C136IC@IFS; Japan: Tokyo Overnight Call Rate, C158IM@IFS; United Kingdom: Official Bank Rate, N112RTAR@G10; United States: Federal Funds Rate, BI 11GDPC@DAILY; Australia: Official Cash Rate, N193RTAR@G10; Austria: Discount Rate, C122IC@IFS; Belgium: 3-month Interbank Rate, C124IM@IFS; Finland: Discount Rate, C172IFC@IFS; Greece: Central Bank Rate, C174IC@IFS; Ireland: short-term facility rate, C178IC@IFS; Netherlands: Discount Rate (prior 1993Q4) C138IC@IFS, Inter Bank Offer Rate (94Q1-98Q4) C138FRIO@IFS; Norway: Discount Rate, C142IC@IFS; Portugal: Discount Rate, C182IC@IFS; South Korea: Discount Rate, C542IFC@IFS; Spain: Central Bank Rate, C184IC@IFS; Sweden: Overnight Money Rate, C144FRUO@ OECDMEI; Switzerland: Discount Rate, B146IC@IFS; Eurozone (post 1999Q1): ECB main refinancing Rate, N023RTAR@G10.

Price Index. Canada: CPI (prior 1961Q1), C156CZN@ OECDMEI, Core CPI, C156CZCN@OECDMEI; France: CPI(prior to 1970Q1), C132CZN @OECDMEI, Core CPI, C1 32CZCN@OECDMEI; Germany: CPI (prior to 1962Q1), C1 34CZN@OECDMEI, Core CPI, C134CZCN@OECDMEI; Italy: CPI (prior to 1960Q1), C136CZN@OECDMEI. Core CPI, C136CZCN@OECDMEI; Japan: Core CPI, C134 CZCN@OECDMEI; United Kingdom: CPI (prior to 1970Q1), C112CZN@OECDMEI, Core CPI, C112CZ CN@OECDMEI; United States: Core CPI, SHIP CXG@G10; Australia: CPI (prior to 1976Q3), C193 CZN@OECDMEI, Core CPI, C193CZCN@OECDMEI; Austria: CPI (prior to 1966Q1), C122CZN@OECDMEI, Core CPI, C122CZCN@OECDMEI Belgium: CPI (prior to 1976Q2), C124CZN@OECDMEI. Core CPI, C124CZCN@OECDMEI; Finland: Core CPI, C172CZC N@OECDMEI; Greece: CPI (prior to 1970Q1), C174CZN @ OECDMEI, Core CPI. C174CZCN@OEC DMEI; Ireland: CPI (prior to 1975Q4), Central Statistics Office of Ireland, Core CPI, C178CZCN@OECDMEI; Netherlands: CPI(prior 1960Q2), C138PC@IFS, Core CPI, C138CZCN@OECDMEI; Norway: CPI(prior 1979Q1), C142CZN @OECDMEI, Core CPI, C142CZCN @OECDMEI; Portugal: CPI (prior 88Q1), C182CZN @ OECDMEI, Core CPI, C182CZCN @ OECDMEI; South Korea (prior to 1990Q1): C542CZN@OECD MEI, Core CPI. C542CZCN@OECDMEI; Spain: CPI (prior to 1976Q1), C184CZN@OECDMEI, Core CPI, C184CZCN@OECDMEI; Sweden: CPI (prior to 1970Q1), C144CZN@OECDMEI, Core CPI, C144CZCN@OECD MEI; Switzerland: Core CPI, C146CZCN@OECDMEI. (17)

Population. Canada: C156TB@UNPOP; France: C132TB@UNPOP: Germany: C134TB@UNPOP; Italy: C136TB@UNPOP; Japan: C158TB@UNPOP; United Kingdom: C112TB@UNPOP; United States: C111TB@UN POP; Australia: C193TB@UNPOP; Austria: C122TB@UN POP; Belgium: C124TB@UNPOP; Finland: C172TB@UN POP; Greece: C174TB@UNPOP; Ireland: C178TB@UN POP; Netherlands: C138TB@UNPOP; Norway: C142TB@ UNPOP; Portugal: C182TB@UNPOP; South Korea: C542TB@UNPOP; Spain: C184TB@UNPOP; Sweden: C144TB@UNPOP; Switzerland: C146TB@UNPOP.

Datafor BRICS

Real GDP. The OECD Quarterly National Accounts database in HAVER includes estimates of PPP-adjusted real GDP in levels for Brazil, India, and South Africa: Brazil: C223GDPC@OECDNAQ (from 1996Q1); India: G534GDPC@OECDNAQ (from 1996Q2); South Africa: E199GDPC@OECDNAQ (from 1960Q1). PPP-adjusted real GDP data in USD for Russia and China are not available directly in the OECDNAQ database. We construct these series using each country's share of global GDP based on PPP as reported in the IMFWEO database, specifically, series A922GPPS@ IMFWEO (Russia) and A924GPPS@ IMFWEO (China) and then use each country's GDP share in 2011 to back out its real GDP in 2011 and infer the whole GDP series based on their real GDP denominated in local currency.

Interest Rate. Brazil: Federal Funds Rate, C223FRAD@0 ECDMEI; Russia: Discount Rate, C922FROS@OECDMEI; India: Discount Rate, C534IFC@IFS; China: Discount Rate, C924IFC@IFS; South Africa: Discount Rate, C199IC@IFS.

Price Index. Brazil: Core CPI, C223PCX@EMERGELA; Russia: CPI excl food, C922CGFN@OECDMEI (prior 2003Q1), Core CPI, H922PCX@EMERGECW; India: CPI (prior to 2006Q1), C534CZN@OECDMEI, Core CPI. N534PCXG @EMERGEPR; China: CPI (prior to 2005Q1), H924PC@EMERGE, CoreCPI, H924PCXZ@EMERGEPR; South Africa: CPI (prior to 2002Q1), H199PC@EMERGE, Core CPI, N199PCXG@ EMERGE.

ABBREVIATIONS

AE: Advanced Economies

AR: Autoregression

BRICS: Brazil, Russia, India, China and (sometimes) South Africa

CPI: Consumer Price Index

GDP: Gross Domestic Product

GLS: Generalized Least Squares

IMF: International Monetary Fund

IS: Investment/Saving

LW: Laubach-Williams

OECD: Organisation for Economic Co-operation and Development

PPP: Purchasing Power Parity

QE: Quantitative Easing

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(1.) See, for example, Laubach and Williams (2003), Clark and Kozicki (2005), Berger and Kempa (2014), Barsky, Justiniano, and Melosi (2014), Curdia et al. (2015), Hamilton et al. (2016), Pescatori and Turunen (2015), and Holston, Laubach, and Williams (2017).

(2.) The 20 advanced countries include Canada, France, Germany, Italy, Japan, United Kingdom, United States, Australia, Austria, Belgium, Finland, Greece, Ireland, Netherlands, Norway, Portugal, South Korea, Spain, Sweden, and Switzerland. We limit our definition of the "world" to an aggregate of these 20 countries due mainly to data availability issues, although in our robustness analysis we consider the implications of including the BRICS countries (Brazil, Russia, India, China, South Africa) in the aggregate for the shorter time period for which data are available for these countries.

(3.) For more detailed discussion on the "pile-up problem," see Stock (1994), Stock and Watson (1998) among others.

(4.) To proceed the Kaiman filter/smoother procedure, we need to set the conditional expectation and covariance matrix of initial states. In both steps, different from Laubach and Williams (2003), the conditional expectation and covariance matrix of initial states are set by diffuse prior instead of the GLS method introduced by Harvey (1989). There are two reasons for the deviation. First, as is mentioned by Laubach (2002), the GLS method tends to exacerbate the "pile-up problem." Second, as in Laubach and Williams (2003), the GLS method fails in the last step because of singularity problems. Thus, it is more consistent to use diffuse prior in both steps rather than using GLS method in the first step while using a diffuse prior in the second step.

(5.) This approach to measuring the world nominal interest rate as a GDP-weighted average of individual national interest rates is very much in the spirit of King and Low (2014).

(6.) Due to data availability, before 1981 we use a fixed window of the data from 1959 to 1981 to estimate the coefficients of the AR model. After 1981 the AR coefficients are estimated using a rolling window with the sample size fixed at 80 quarters

(7.) The U.S. natural rate estimated by the Laubach-Williams model is also updated in real time on the website of the San Francisco Fed. Our replication matches with their results closely. The slight difference might arise as a result of the different observation vintage. Our data are observed in June 2016 as our world natural rate estimates while the data used by Laubach and Williams for the same sample period are observed in March 2016.

(8.) The estimates reported by Laubach and Williams (2003) using a shorter sample of U.S. data range from 0.032 to 0.089.

(9.) Earlier studies of the impact of openness on the Phillips curve estimates include Tootell (1998), Gamber and Hung (2001), Razin and Yuen (2002), and Balakrishnan and Ouliaris (2006).

(10.) We use the OECD business cycle chronology for two reasons. First, our sample of countries is all OECD members and the aggregate of the 20 countries we select makes up dominant share of the total GDP of OECD countries. Second, it is the only public source we are aware of that dates the turning points of global economic activity back to the 1960s. Martinez-Garcia, Grossman, and Mack (2015) provide a global business cycle chronology for a broader group of countries but their chronology only begins in 1980.

(11.) Here we assume that this demographic shift is exogenous.

(12.) According to estimates reported in the April 2017 edition of the IMF's World Economic Outlook.

(13.) While data are available for the BRICS countries from about the mid-1990s on, we elected to start our sample in 1999 so as to avoid the high inflation episode in Russia and Brazil in the first half of the 1990s. In Russia inflation was running at rates in excess of 200%. In Brazil, inflation peaked at 4,922% in June 1994, and the Central Bank of Brazil's main policy rate reached 132,532%. A second reason for starting in 1999 rather than earlier is that this date is closer to the date of China's accession to the World Trade Organization.

(14.) The GDP share is time-varying as is depicted in Figure 1. It is the ratio between the PPP-adjusted real GDP and the aggregated real GDP of the 20 countries.

(15.) Specifically, Canada, Italy, United Kingdom, United States, Australia, Austria, Finland, Greece, Netherlands, Norway, Portugal, South Korea, Spain, and Switzerland.

(16.) The CPI of Ireland before 1975Q4 is acquired from the Central Statistics Office of Ireland.

(17.) Except for the United States, we import the nonseasonal-adjusted price series from the Haver since they have longer samples. Then we make the seasonal adjustment to the data using Haver built in function. The early Ireland CPI data are seasonally adjusted by Tramo-Seats.

MARK A. WYNNE and REN ZHANG *

* The views in this article are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.

Wynne: Vice President, Research Department. Federal Reserve Bank of Dallas, Dallas, TX 75201. Phone 214-922-5159, Fax 214-922-5194, E-mail mark.a.wynne@dal.frb.org

Zhang: Assistant Professor, Department of Economics, Bowling Green State University, Bowling Green, OH 43403. Phone 469-394-3198, Fax 419-372-1557, E-mail renz@bgsu.edu

doi: 10.1111/ecin.12500

Online Early publication September 22, 2017

Caption: FIGURE 1 GDP Share: 1960Q1-2015Q4

Caption: FIGURE 2 The World Output Gap: 1961Q1-2015Q4

Caption: FIGURE 3 The World Potential Output Growth and Its Trend (Annualized)

Caption: FIGURE 4 The World Real Interest Rate and Natural Rate of Interest

Caption: FIGURE 5 The World Natural Rate and Its Decomposition

Caption: FIGURE 6 The U.S. Natural Rate and Its Decomposition

Caption: FIGURE 7 The World Natural Rate Estimates with Different Value of Parameter c

Caption: FIGURE 8 Aggregated World Population Growth Rate

Caption: FIGURE 9 The World Real Interest Rate and Natural Rate of Interest (Per Capita)

Caption: FIGURE 10 The World Real Interest Rate and Natural Rate of Interest (Including BRICS)

Caption: FIGURE 11 Comparing Different Estimates of the Natural Interest Rate

TABLE 1
Model Parameter Estimates

Parameters             Baseline           LW

[a.sub.1]            1.554 (14.56)   1.553 (14.61)
[a.sub.2]            -0.632 (5.88)   -0.598 (5.71)
[a.sub.3]            -0.035 (1.93)   -0.058 (3.18)
[b.sub.1]            0.782 (10.94)   0.569 (8.52)
[b.sub.2]            0.114 (1.35)    0.379 (4.34)
[b.sub.3]            0.159 (2.45)    0.040 (1.36)
[b.sub.4]            0.002 (1.81)    0.0025 (2.18)
[b.sub.5]                 --         0.036 (3.38)
c                    0.458 (1.14)    1.321 (2.22)
[[sigma].sub.1]          0.343           0.360
[[sigma].sub.2]          0.706           0.767
[[sigma].sub.3] =        0.127           0.248
  [[lambda].sub.z]
  [[sigma].sub.1]/
  [a.sub.3]
[[sigma].sub.4]          0.229           0.599
[[sigma].sub.5]          0.171           0.042
[[sigma].sub.r] =        0.149           0.254
  [square root of
  ([c.sup.2]
  [[sigma].sup.2
  .sub.5] +
  [[sigma].sup.2
  .sub.3])]
[[lambda].sub.z]         0.013           0.040
[[lambda].sub.g]         0.187           0.017

Notes: t-Statistics are reported in parentheses.
LW, Laubach-Williams.

TABLE 2
Model Parameter Estimates: Robustness Check I

Parameters             Baseline          c = 1         c = 1.321

[a.sub.1]            1.554 (14.56)   1.543 (14.18)   1.542 (14.10)
[a.sub.2]            -0.632 (5.88)   -0.621 (5.64)   -0.619 (5.58)
[a.sub.3]            -0.035 (1.93)   -0.023 (1.67)   -0.017 (1.48)
[b.sub.1]            0.782 (10.94)   0.781 (10.90)   0.781 (10.91)
[b.sub.2]            0.114 (1.35)    0.116 (1.37)    0.117 (1.38)
[b.sub.3]            0.159 (2.45)    0.161 (2.43)    0.162 (2.43)
[b.sub.4]            0.002 (1.81)    0.002 (1.89)    0.002 (1.89)
[b.sub.5]                 --              --              --
c                    0.458 (1.14)          1             1.321
[[sigma].sub.1]          0.343           0.350           0.352
[[sigma].sub.2]          0.706           0.706           0.706
[[sigma].sub.3] =        0.127           0.198           0.269
  [[lambda].sub.z]
  [[sigma].sub.1]/
  [a.sub.3]
[[sigma].sub.4]          0.229           0.222           0.220
[[sigma].sub.5]          0.171           0.173           0.174
[[sigma].sub.r] =        0.149           0.263           0.354
  [square root of
  [c.sup.2]
  [[sigma].sup.2
  .sub.5] +
  [[sigma].sup.2
  .sub.3]]
[[lambda].sub.z]         0.013           0.013           0.013
[[lambda].sub.g]         0.187           0.195           0.198

Note: t-Statistics are reported in parentheses.

TABLE 3
Model Parameter Estimates: Robustness Check II

Parameters             Baseline       Per Capita      With BRICS

[a.sub.1]            1.554 (14.56)   1.569 (15.41)   1.406 (12.66)
[a.sub.2]            -0.632 (5.88)   -0.653 (6.24)   -0.649 (5.47)
[a.sub.3]            -0.035 (1.93)   -0.034 (1.94)   0.170 (1.87)
[b.sub.1]            0.782 (10.94)   0.763 (9.91)    0.236 (2.99)
[b.sub.2]            0.114 (1.35)    0.123 (1.44)    0.311 (3.43)
[b.sub.3]            0.159 (2.45)    0.186 (2.25)    0.103 (1.51)
[b.sub.4]            0.002 (1.81)    0.002 (1.84)    0.001 (0.80)
[b.sub.5]                 --              --              --
c                    0.458 (1.14)    0.600 (1.21)    1.466 (3.58)
[[sigma].sub.1]          0.343           0.330           0.339
[[sigma].sub.2]          0.706           0.700           0.515
[[sigma].sub.3] =        0.127           0.116           0.517
  [[lambda].sub.z]
  [[sigma].sub.1]/
  [a.sub.3]
[[sigma].sub.4]          0.229           0.243           0.052
[[sigma].sub.5]          0.171           0.168           0.009
[[sigma].sub.r] =        0.149           0.154           0.517
  [square root of
  ([c.sup.2]
  [[sigma].sup.2
  .sub.5] +
  [[sigma].sup.2
  .sub.3])]
[[lambda].sub.z]         0.013           0.012           0.259
[[lambda].sub.g]         0.187           0.174           0.045

Notes: t-Statistics are reported in parentheses.
LW, Laubach-Williams.