Abstract

Asset pricing models predict a strong connection between the real risk-free interest rate and the macroeconomy, but prior research finds little empirical support for the connection when examining expected growth. This paper documents a robust relation between the interest rate and macroeconomic uncertainty (i.e., conditional variance). Consistent with precautionary savings, high uncertainty is associated with a low interest rate using numerous data sources, time periods, and measures. A relation between habit and the interest rate disappears after including uncertainty, and the relation is stronger using long-run uncertainty. The results imply that analyses of the interest rate without uncertainty are seriously incomplete.

Received September 17, 2014; accepted January 8, 2016 by Editor Jeffrey Pontiff.

In the concrete world, the most conspicuous characteristic of the future is its uncertainty.

—Irving Fisher, The Rate of Interest: Its Nature, Determination and Relation to Economic Phenomena

Recent events underscore the importance of understanding how fluctuations in macroeconomic uncertainty interact with asset markets. While uncertainty about future economic growth is thought to have a broad impact on the economy, empirically and theoretically identifying its influence is difficult to do cleanly. An exception to this, at least in theory, is the interest rate. A central prediction of finance (with roots as early as Fisher 1907) is that the real risk-free interest rate is determined by two economic forces. Intertemporal smoothing induces a positive relation between the interest rate and expectations of economic growth, and precautionary savings induces a negative relation between the interest rate and uncertainty, the conditional variance of growth. This relation describes an important link between uncertainty and a key economic variable that can be estimated with minimal assumptions. However, there is currently little empirical evidence for the direct link between the risk-free rate and macroeconomic uncertainty.

This paper examines this fundamental relation and finds that, consistent with the precautionary savings motive, there is an economically and statistically strong negative relation between the real interest rate and uncertainty. When regressing the real annualized three-month Treasury-bill rate on annual estimates of growth and uncertainty, a one-standard-deviation increase in uncertainty is associated with a 1.2% to 2.3% decrease in the level of the risk-free rate when growth is measured by consumption, GDP, or industrial production. In each case these coefficients have a t-statistic greater than three (in absolute value). The adjusted R2 of the model increases from roughly zero when using expected growth alone to 14%, 55%, and 35% when both uncertainty and growth are included in the regression.

As this pattern arises in a number of models and applications, I do not examine it within the context of a specific model. The main contribution of this paper is to provide robust, reduced-form empirical evidence of the link between economic uncertainty and the interest rate. Indeed, the analysis in this paper is new in that it focuses on the broad pattern implied by precautionary savings and ignores the specific cross-equation restrictions implied by various models. While this limits the implications for a specific stochastic discount factor, it provides robust evidence for the negative relation between the interest rate and uncertainty without the limitation of jointly testing the numerous conflicting restrictions imposed by each of the many models. By applying a reduced-form approach, along with a number of different econometric specifications, data sources, and time periods, the paper yields a deeper understanding of a central economic variable, the interest rate, and how the interest rate is linked to the macroeconomy.

The intertemporal smoothing and precautionary savings motives described by these theories are based on expectations of growth and uncertainty, so the measurement of these variables is key to the analysis. The baseline estimates described above are from an ARMA-GARCH model fit in sample. Annual macroeconomic data are used as it is available for a long and consistent time series beginning in 1934. While this analysis offers a natural starting point for the empirical exercise, there are inherent limitations. To demonstrate the robustness of the results, I explore other data sources and methodologies.

One possible worry is that the time series of data is too short and does not allow enough information to properly estimate the coefficients. I extend the analysis to data covering 140 years by piecing together data from various sources. Using this long time series, an economically and statistically significant negative relation between growth and uncertainty is found. Using a quarterly time series of macroeconomic data covering a shorter time horizon, but with more frequent observations, also yields a negative and significant relation between uncertainty and the interest rate. The main analysis utilizes the three-month Treasury-bill rate, but the results also hold for the 30-day, 6-month, and 1-year interest rates.

The United States may be unique either due to the realization of random shocks in a single time series or because of its special position in international asset markets. One interpretation of the negative coefficient on uncertainty is that in uncertain times there is a flight to quality, where money floods into the United States. This would depress the interest rate in the United States and increase the interest rate abroad. International data from eight developed economies—Belgium, Canada, France, Germany, Italy, Japan, the Netherlands, and the United Kingdom—are examined. When measuring growth using consumption or GDP, nearly all of the countries display an inverse relation between the interest rate and uncertainty. This is not consistent with the negative coefficient on the United States relation being caused by a flight to quality to the United States from these countries.

Forecasts from the time-series model fit in sample, using the entire sample, may not accurately reflect the ex ante variables in the model, either due to weak forecasts or to a look-ahead bias. Weak growth forecasts could lead to the spurious finding of a negative coefficient on the uncertainty term or the finding of no relation between expected growth and the interest rate. Using ex post realized growth instead of estimates from a time-series model yields similar results, as do rolling time-series estimates using only data known at the time the forecast is made. Ex ante forecasts from the Survey of Professional Forecasters also yield a negative coefficient on uncertainty.

Perhaps the most commonly referenced ex ante forecaster of uncertainty is the VIX index. While only available for a short time period, the VIX is used to proxy for uncertainty, while both the Leading Index of the United States and the Consumer Sentiment Index are used for monthly growth forecasts. These measures are used in practice to forecast economic activity and do not utilize the time-series methods employed elsewhere in the paper. Testing the relation with these measures yields a negative and significant association between the risk-free rate and uncertainty. The VIX is meant to forecast stock market volatility, which can be directly calculated for a much longer time series. Using various measures of market volatility over this extended time period, I find a negative and significant relation with the risk-free interest rate.

I test empirical specifications, suggested by the model for nominal interest rates, that include the variance of inflation, along with covariance terms between inflation and uncertainty. Inflation risk is measured using a time-series model, as well as by examining the dispersion of expert forecasts. Controlling for these additional factors, the relation between growth and uncertainty remains, suggesting that inflation risk is not responsible for the relation.

Econometric issues related to model choice and measurement error are examined, and the results are shown to be robust to many possible concerns. The pattern is not driven by outliers or model choice regarding the number of lags in the ARMA process. The results are robust to using longer sampling intervals and to econometric corrections for persistence in the regressors.

The early empirical work analyzing the risk-free rate and the macroeconomy presented a puzzle as it found no empirical support for a link between the two. This was because these studies assumed that there was no time-varying uncertainty and as such only examined the relation between the real interest rate and forecasts of growth (e.g., Hall 1978). While some attempt was made to reconcile the lack of an empirical relation between growth and uncertainty with the theory (e.g., Campbell and Mankiw 1989), a puzzle remained as to why the interest rate and the macroeconomy appear empirically unrelated.

This paper helps to partially resolve the puzzle. First, the strong link between uncertainty and the risk-free rate shows that the theory has explanatory power, but that the uncertainty channel is the strongest. Second, coefficients on growth for the United States are not significantly different from zero, but they are also not significantly different from the small positive coefficients predicted by the long-run risks model. It may be that uncertainty is more precisely measured than the expectation of growth (Merton 1980), and the measurement error associated with growth could account for the lack of a significant empirical relation (Bansal and Yaron 2004; Bansal, Kiku and Yaron 2012). Finally, extending the analysis to international data, a number of countries display both a significant inverse relation between the interest rate and uncertainty and a positive relation between the interest rate and growth. Thus, the data support the importance of an economic link between the risk-free rate and the macroeconomy.

The results are based on a minimum of assumptions and thus are relevant to a number of models. For example, a broad class of models, based on Campbell and Cochrane (1999), examine the impact of time-varying risk aversion linked to external habit. The original model utilizes i.i.d. consumption growth and a constant risk-free rate, but subsequent papers have relaxed this assumption, yielding interest rates that vary with habit. This adds additional terms to the interest rate process that could account for the relation between uncertainty and the interest rate. Further, there has been debate about the cyclicality of the risk-free rate in these models: Verdelhan (2010) requires a procyclical interest rate to explain currency markets, while Wachter (2006) argues for a countercyclical interest rate to account for the domestic bond market. Without controlling for uncertainty, habit has a negative and significant relation to the risk-free rate, but after uncertainty is accounted for, the relation between habit and the interest rate is weak and insignificant. After controlling for habit, the coefficient on uncertainty is negative, significant, and roughly unchanged. The evidence is consistent with the risk-free rate having a strong relation with economic uncertainty, but not with time-varying risk aversion.

The long-run risks model (Bansal and Yaron 2004) is one of the few models to include time-varying uncertainty. In this model, the precautionary savings motive is based on long-run, rather than short-run, uncertainty. Empirically, both long-run and short-run measures of uncertainty yield negative and significant relations with the risk-free rate. After including both measures of uncertainty in a regression, the short-run measure becomes insignificant, while the long-run measure remains large, negative, and significant.

The central contribution of this paper is to provide empirical support and a description of the fundamental relation between economic uncertainty and the interest rate. While there is a large literature exploring and modeling the properties of the interest rate, there is little research exploring the economic causes of real interest rate behavior.1 Further, by focusing on the relation between growth and the interest rate, the literature examining the economics of the real interest rate has found little support for the theory. This paper shows that while growth and the interest rate have a noisy empirical relation, the relation between uncertainty and the risk-free rate is large, lending strong support to the basic finance theory.

This paper contributes to the growing literature exploring how macorecoomic uncertainty intersects with financial markets. For example, Bansal and Shaliastovich (2013) provide evidence of a strong link between uncertainty and the bond risk premium. A number of recent works provide direct evidence of a link between uncertainty and equity markets. Bansal, Khatchatrian, and Yaron (2005) and Nakamura, Sergeyev, and Steinsson (2014) show that increases in macroeconomic uncertainty decrease aggregate equity prices. Bansal et al. (2014) and Boguth and Kuehn (2013) examine the link between uncertainty and risk premium.

This paper also contributes to the literature on the impact of uncertainty in the economy. Bloom (2009) examines firm-level data and finds that shocks to uncertainty correspond to rapid drops and rebounds in employment and output. Bloom, Floetotto, and Jaimovich (2010) study the impact of uncertainty on the business cycle and find that increases in uncertainty lead to drops in economic activity. Justiniano and Primiceri (2008), who examine a dynamic stochastic general equilibrium model with parameter uncertainty, find the same. This paper adds further evidence that uncertainty is an important economic fundamental with wide-ranging impact.

1. A Basic Model

A number of asset pricing models imply a simple linear relation between the real interest rate, growth, and uncertainty of the form:  

(1)
rt=β0+β1Etgt+1+β2Vartgt+1,
where rt is the log of the time t to t+1 risk-free rate, gt+1 is the log consumption growth rate in the subsequent period, Et is the expectation conditioning on information at time t, Vart is the variance conditioning on information at time t, and β0, β1, and β2 are constant coefficients. The ex ante variance of growth is what I term uncertainty.

Of course, Equation (1) could be, and has been, tested by imposing restrictions on the β coefficients implied by various models, but each model implies different restrictions on the beta coefficients. Appendix A shows that the simple discrete-time representative agent model with conditionally lognormal disturbances (as in Hansen and Singleton 1983, Ferson 1983, and Harvey 1988), along with a version of habit formation (Campbell and Cochrane 1999) and long-run risks (Bansal and Yaron 2004), yield linear combinations of growth and uncertainty that can be expressed using Equation (1). While the precise coefficients differ for the various models, they each imply β1>0 and β2<0. This general prediction is the focus of my analysis.

In choosing to examine the broad prediction made by a number of models, I am ignoring the cross-equation restrictions implied by each of these models. For example, the classic lognormal model tested by Hansen and Singleton (1983) suggests that β1 is equal to the coefficient of risk aversion, and β2 is equal negative one-half of risk aversion squared. They, and the many papers that followed, reject the model as it is not possible to empirically satisfy these restrictions with the other cross-equation restrictions implied by the model.

This paper instead focuses on the economic intuition that precautionary savings should induce a negative relation between the risk-free rate and macroeconomic uncertainty. The shortcoming of this approach is that its direct implications for a specific model are limited. The strength is that by examining a broad economic prediction made by a number of models, this paper provides direct empirical evidence of a strong link between the real risk-free interest rate and macroeconomic uncertainty without imposing the numerous and often conflicting restrictions assumed by each of the separate models. The results are economically meaningful not in their specific implications for a stochastic discount factor but in demonstrating that shifts in uncertainty are historically associated with large changes in the interest rate.

2. Data and Summary Statistics

Unless otherwise noted, growth is measured at an annual rate to avoid issues with seasonal adjustment and extend the analysis to as many years as possible. All measures of growth end in 2010. Growth is measured as the log of the ratio of the level measure (consumption expenditures, GDP, and output) at time t and the growth measure at time t-1. All growth measures are in per capita real dollars.

This paper uses three different measures of economic growth. One measure is real consumption growth from the National Income and Product Accounts (NIPA), where consumption is defined as nondurable goods, plus services. Additionally, the paper uses Real NIPA GDP the Federal Reserve’s G.17 real industrial production index, as an alternate measure of growth. Consumption and GDP growth data begin in 1930, and data from the Industrial Production index begin in 1920.

To extend the analysis, the above series is augmented with historical data. For consumption, the Kuznets-Kendricks series (Kuznets 1961; Kendrick 1961), compiled by Shiller (1982, 1989) and starting in 1890, is used. Romer has argued that the methodology used to compile these data accentuates fluctuations, making it more volatile than the economy actually was. Chapman (1998, 2002) and Otrok, Ravikumar, and Whiteman (2002) econometrically examine the time series of consumption and find similar evidence of structural breaks in the data. Unfortunately, there is not an alternative consumption series, but Romer (1989) has compiled an alternative GNP series starting from 1871. An industrial production index, compiled by Miron and Romer (1990) and extending to 1885, is used. Results for quarterly growth are presented as well, where growth is measured as one, plus the log of current level, divided by the level four quarters previous. The quarterly data begin in 1947.

Nominal interest rates are measured as the log of one, plus the decimal fraction interest rate. An annual rate is not available for the entire period of analysis, so the rate is calculated by annualizing the three-month December rate, which is the last observation before the period of the growth forecast.2 The main interest rate used is the three-month Treasury-bill rate from the Federal Reserve Bank of St. Louis (TB3MS series), converted from a bank discount basis to an effective yield basis. This series is available starting in 1934. An alternative measure of the interest rate is the one-month Treasury-bill rate from Ibbotson Associates, compiled by Fama and French. The first full year of data is available for 1927. The one-year interest rate is the GS1 series starting in 1954, and that from 1947 to 1953 comes from Homer and Sylla (2005). A historical series of the six-month rate (again, the December rate from the previous year annualized), which reaches back to 1871, is used. This series is compiled using the 4- to 6-month prime commercial paper rate from Macauly (1938) from 1869 to 1937, the Federal Reserve from 1938 to 1970, the six-month commercial paper rate from 1971 to 1997, and the six-month CD rate from 1997 to 2010.3

In the analysis, the interest rates are net of expected inflation. Annual inflation is calculated as the log of CPI in December of year t, divided by CPI in December of year t-1. This is modeled using an ARMA(1,1) process, and the predicted value is used as the estimate of expected inflation (as in Constantinides and Gosh 2011).4 This is subtracted from the log of the interest rate to give the expected real interest rate. Section 4.5 analyzes whether the assumptions necessary for this specification are supported in the data.

Table 1 presents summary statistics. All of the measures of economic growth are positively correlated, with GDP and consumption having a correlation of 0.53, industrial production and consumption having a correlation of 0.47, and industrial production and GDP having a correlation of 0.86. GDP and industrial production growth are more volatile than consumption growth, with standard deviations more than twice as large. Also, consistent with the Romer critique of the Kuznets series, consumption prior to 1930 is much more volatile than consumption after, with double the standard deviation.

Table 1

Summary statistics

A. Measures of growth rates 
 Mean First year Last year SD Min 25th pctile Median 75th pctile Max 
Consumption 0.018 1930 2010 0.022 −0.080 0.010 0.021 0.031 0.073 
GDP 0.021 1930 2010 0.049 −0.146 0.002 0.023 0.043 0.158 
Industrial production 0.020 1920 2010 0.096 −0.282 −0.008 0.027 0.069 0.228 
Historic consumption 0.020 1890 1929 0.044 −0.085 −0.010 0.018 0.050 0.099 
Historic GDP 0.038 1871 1929 0.035 −0.043 0.014 0.038 0.058 0.152 
Historic industrial production 0.038 1885 1940 0.127 −0.351 −0.015 0.046 0.123 0.374 
 
B. Nominal annualized interest rates 
 
 Mean First year Last year SD Min 25th pctile Median 75th pctile Max 
 
3 month 0.040 1935 2010 0.032 0.000 0.012 0.037 0.057 0.160 
30 day 0.036 1927 2010 0.031 0.000 0.010 0.033 0.054 0.156 
6 month 0.049 1871 2010 0.028 0.003 0.032 0.049 0.061 0.172 
1 year 0.052 1947 2010 0.033 0.004 0.028 0.049 0.073 0.161 
A. Measures of growth rates 
 Mean First year Last year SD Min 25th pctile Median 75th pctile Max 
Consumption 0.018 1930 2010 0.022 −0.080 0.010 0.021 0.031 0.073 
GDP 0.021 1930 2010 0.049 −0.146 0.002 0.023 0.043 0.158 
Industrial production 0.020 1920 2010 0.096 −0.282 −0.008 0.027 0.069 0.228 
Historic consumption 0.020 1890 1929 0.044 −0.085 −0.010 0.018 0.050 0.099 
Historic GDP 0.038 1871 1929 0.035 −0.043 0.014 0.038 0.058 0.152 
Historic industrial production 0.038 1885 1940 0.127 −0.351 −0.015 0.046 0.123 0.374 
 
B. Nominal annualized interest rates 
 
 Mean First year Last year SD Min 25th pctile Median 75th pctile Max 
 
3 month 0.040 1935 2010 0.032 0.000 0.012 0.037 0.057 0.160 
30 day 0.036 1927 2010 0.031 0.000 0.010 0.033 0.054 0.156 
6 month 0.049 1871 2010 0.028 0.003 0.032 0.049 0.061 0.172 
1 year 0.052 1947 2010 0.033 0.004 0.028 0.049 0.073 0.161 

This table presents summary statistics for the data analyzed in the paper. Panel A presents summary statistics for the various measures of the log of one, plus economic growth, and panel B presents summary statistics of the log of nominal interest rates.

The Survey of Professional Forecasters, conducted by the Federal Reserve Bank of Philadelphia, provides an alternative forecast of growth, presumably made using an information set richer than simple lagged values of growth. The survey is timed to coincide with the release of the advanced estimates from the previous quarter, so the forecast of growth is made in the first quarter of the year when the value for the fourth quarter from the previous year is known. Thus, the growth estimate is the log of the forecast for the fourth quarter of year, t+1, divided by the known value of the fourth quarter in year t. The time series of data is relatively short with real consumption growth calculated from 1981 and GDP and industrial production calculated from 1968. Forecasts of inflation are calculated in the same manner as forecasts of growth using forecasts of the GDP price index from 1968.

Results using ex ante forecast data are presented using the VIX to proxy for uncertainty, and the Leading Index for the United States and the University of Michigan Consumer Sentiment Index to proxy for growth. Each variable is analyzed in logs. The VIX index begins in 1990 and is constructed using options data to measure the thirty-day expected volatility of the S&P 500.5 To deal with the short time series, I utilize growth forecasts that are available monthly. The leading index corresponds to a six-month forecast, constructed using forward indicators, such as housing permits, unemployment insurance claims, among others. The Consumer Sentiment Index is based on a monthly telephone survey meant to gauge consumer confidence.6 A related measure to the VIX available for a longer time period is stock market volatility. Market volatility is calculated as the variance of daily returns measured by the CRSP value-weighted return index.

Finally, I use data on eight international developed countries, namely, Australia, Belgium, Canada, France, Italy, Japan, the Netherlands, and the United Kingdom. These countries have available a long time series of data. Consumption and growth are from Barro and Ursúa (2008), and the three-month Treasury-bill rate and CPI for each country are from Global Financial Data.7

3. Empirical Results

3.1 Baseline results

To analyze the linear relation from Equation (1), estimates of expected economic growth and the variance of economic growth are needed. As a baseline case, I model growth as a time-series process. The most basic formulation is to model growth as ARMA(1,1):8 

(2)
gt+1=ϕgt+θϵt+μ+ϵt+1.

After estimating the model, it is used to predict ĝt+1, which is then used as the estimate for Et[gt+1] in Equation (1). The simplest way to estimate Vart[gt+1] is as the square of the residuals in period t, so the estimate of the variance is t -gt)2.Table 2, panel A, takes the estimates from this model to estimate Equation (1).

Table 2

Regressions of real interest rate on forecasts of growth and uncertainty

A. Residual methodology 
 Consumption GDP Industrial production 
Et[gt+1−0.573 −0.127 −0.429 
 (−0.75) (−0.91) (−1.43) 
Vart[gt+1−23.906 −8.161 −1.002 
 (−3.01) (−8.14) (−1.98) 
Constant 0.024 0.020 0.023 
 (1.47) (3.36) (2.44) 
 
Observations 75 75 75 
Adjusted R2 0.08 0.42 0.15 
R2 excluding Vart[gt+1−0.01 −0.01 0.00 
Impact of 1 SD Vart[gt+1−0.010 −0.020 −0.012 
 
B. GARCH methodology 
 
 Consumption GDP Industrial production 
 
Et[gt+1−0.834 0.239 0.049 
 (−0.88) (1.14) (0.20) 
Vart[gt+1−60.389 −12.902 −2.441 
 (−3.30) (−11.22) (−3.04) 
Constant 0.038 0.020 0.020 
 (1.72) (3.03) (3.72) 
 
Observations 76 76 76 
Adjusted R2 0.14 0.55 0.35 
R2 excluding Vart[gt+10.00 −0.01 0.00 
Impact of 1 SD Vart[gt+1−0.012 −0.023 −0.018 
A. Residual methodology 
 Consumption GDP Industrial production 
Et[gt+1−0.573 −0.127 −0.429 
 (−0.75) (−0.91) (−1.43) 
Vart[gt+1−23.906 −8.161 −1.002 
 (−3.01) (−8.14) (−1.98) 
Constant 0.024 0.020 0.023 
 (1.47) (3.36) (2.44) 
 
Observations 75 75 75 
Adjusted R2 0.08 0.42 0.15 
R2 excluding Vart[gt+1−0.01 −0.01 0.00 
Impact of 1 SD Vart[gt+1−0.010 −0.020 −0.012 
 
B. GARCH methodology 
 
 Consumption GDP Industrial production 
 
Et[gt+1−0.834 0.239 0.049 
 (−0.88) (1.14) (0.20) 
Vart[gt+1−60.389 −12.902 −2.441 
 (−3.30) (−11.22) (−3.04) 
Constant 0.038 0.020 0.020 
 (1.72) (3.03) (3.72) 
 
Observations 76 76 76 
Adjusted R2 0.14 0.55 0.35 
R2 excluding Vart[gt+10.00 −0.01 0.00 
Impact of 1 SD Vart[gt+1−0.012 −0.023 −0.018 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. In panel A, forecasts of growth are made from an ARMA(1,1) model, and the variance of growth is the square of the lagged residual. In panel B, forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

To improve the estimate of uncertainty, Table 2, panel B, models growth as a conditionally heteroscedastic process using a GARCH(1,1) model (Engle 1982; Bollerslev 1986). The specification used is  

(3)
gt+1=ϕgt+θϵt+μ+ϵt+1Vartϵt+1=γ+α1ϵt2+α2σt2.

This yields a forecast for growth and uncertainty that the agent can undertake at time t if the agent knows the time-series model of the economy.

Table 2 estimates Equation (1) using annual data on consumption, GDP, and industrial production as measures of growth. For all three measures, and using both methodologies for measuring the variance, the uncertainty term is negative and statistically significant. Further, growth is insignificant in all of the specifications.

Table 2 suggests that there is a strong relation between the interest rate and future economic uncertainty. In examining the final row of Table 2, panel B, “Impact of 1 SD Vart[gt+1],” we see a one-standard-deviation increase in the risk-free rate associated with a 0.012, 0.023, and 0.018 decrease in the interest rate for consumption, GDP, and industrial production, respectively.

Further, the variance adds significant explanatory power to the model over simply including growth. The row labeled “Adjusted R2” gives the adjusted R2 of the estimated model, while the row labeled “R2 excluding Vart[gt+1]” gives the adjusted R2 for the model regressing the interest rate on the growth forecast, while excluding the variance term. Using the residual measure of variance in panel A, the adjusted R2 value significantly increases in all specifications from roughly zero without the uncertainty term to 0.08, 0.42, and 0.15 for consumption, GDP, and industrial production, respectively. Similarly, when using the GARCH specifications, this value moves from roughly zero without uncertainty to 0.14, 0.55, and 0.35 with uncertainty.

Table 2, panel B, contains the main results using, in my opinion, the best blend of econometric specification and data available. Consistent data from the same source exist for the entire period for all measures of the interest rate, inflation, and growth. While the ARMA-GARCH framework has its shortcomings, other options, such as expert forecasts or the VIX, exist only for relatively short time periods and have their own issues. Nevertheless, in most specifications examined in the following sections, the relation between the interest rate and uncertainty remains robust, negative, and economically meaningful.

3.2 The effects of extreme volatility periods

Figure 1 graphs data from the ARMA-GARCH model in Table 2, panel B. Predicted consumption growth is the solid green line, and the risk-free interest rate is the dashed orange line. Sometimes, these lines seem to move together, for example, during the 2000s and surrounding WWII, while during other periods, they follow very different paths, for example, in the early 1980s. There does not appear to be a strong relation between these two series.

Figure 1

Expected consumption growth and three-month Treasury-bill rate

This figure presents the real three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, and forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model.

Figure 1

Expected consumption growth and three-month Treasury-bill rate

This figure presents the real three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, and forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model.

Figure 2 graphs the variance of consumption growth as the solid green line versus the interest rate as the dashed orange line. There are some periods with very high volatility. The periods with the highest volatility occur during major events in U.S. economic history. The first period is the forecasts for 1937 (based on data from 1936), which coincides with the recession within the Great Depression. The next period occurs in the forecasts for 1948, as the United States emerged from World War II. Another year with high volatility is 1975, as the United States dealt with the first OPEC oil embargo. The final high volatility period occurred in 2010, during the recent financial crisis.

Figure 2

Ex ante variance of consumption growth and three-month Treasury-bill rate

This figure presents the real three-month Treasury-bill rate in year t on forecasts of the log of the variance of growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, and forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model.

Figure 2

Ex ante variance of consumption growth and three-month Treasury-bill rate

This figure presents the real three-month Treasury-bill rate in year t on forecasts of the log of the variance of growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, and forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model.

To show the results are not driven by the large relative magnitudes of these periods, the data are winsorized at the 90th percentile, with the results presented in Table 3. All of the results remain robust to this change. The significance of the variance coefficient, adjusted R2, and the impact of a one-standard-deviation change are slightly stronger for consumption and industrial production and slightly weaker for GDP. Table 3 suggests that the results are not driven by outliers in the data.

Table 3

Regressions of real interest rate on forecasts of growth and uncertainty winsorized at the 90th percentile

 Consumption GDP Industrial production 
Et[gt+1−0.681 0.422 −0.146 
 (−0.78) (1.04) (−0.56) 
Vart[gt+1−126.189 −17.249 −3.638 
 (−3.34) (−7.05) (−5.39) 
Constant 0.046 0.019 0.028 
 (2.09) (2.00) (4.40) 
 
Observations 76 76 76 
Adjusted R2 0.17 0.44 0.44 
R2 excluding Vart[gt+1−0.01 0.00 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.020 −0.020 
 Consumption GDP Industrial production 
Et[gt+1−0.681 0.422 −0.146 
 (−0.78) (1.04) (−0.56) 
Vart[gt+1−126.189 −17.249 −3.638 
 (−3.34) (−7.05) (−5.39) 
Constant 0.046 0.019 0.028 
 (2.09) (2.00) (4.40) 
 
Observations 76 76 76 
Adjusted R2 0.17 0.44 0.44 
R2 excluding Vart[gt+1−0.01 0.00 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.020 −0.020 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. Growth and uncertainty forecasts are winsorized at the 90th percentile. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

3.3 The long historical record

The preceding analysis begins with data starting in 1934, the first year that the three-month risk-free rate is available. In this section the analysis is extended to the maximum extent possible. This means using the historic interest rate data, which begins in the late 1800s. GDP, using the Romer series, begins in 1871, and industrial production, using the Miron and Romer estimates, begins in 1885. For consumption, the Kendrick-Kuznets series, which begins in 1890, is used.

Figure 3 graphs the extended Kendrick-Kuznets consumption series with the red vertical line indicating 1930, the first-year NIPA data are used for the growth rate. Prior to the NIPA data, there is significantly more volatility. From 1934 to 2010, the period for which the initial analysis was conducted, the standard deviation is 0.024. Before this period, from 1890 to 1933, the standard deviation is 0.049, more than double the subsequent period. This is not true with the interest rate, which actually has a higher standard deviation in the later period, 0.025 prior to 1934 and 0.31 after. Romer (1989) argues that the higher volatility of the Kendrick-Kuznets series is due to the methodology of its construction, and not actual shifts in the economy. Note that the Romer GNP series is slightly less volatile from 1871 to 1929, than the NIPA series from 1930 to 2010. If the consumption series is artificially noisy, the estimates will be attenuated. To partially offset this, both the standard analysis, as well as analysis in which the estimates of growth and variance are winsorized at the 90th percentile, is presented. Given these data limitations, the estimates based on the historical data using GDP and industrial production probably provide a better representation of the economy than those using consumption. Also, to examine the stability of the coefficients across the sample, the third column for each measure allows for different coefficients on each variable before and after World War II, where Chapman (1998) argues there is a structural break in the data.

Figure 3

Historic consumption growth

This figure graphs the log of one, plus consumption growth from 1890 to 2010.

Figure 3

Historic consumption growth

This figure graphs the log of one, plus consumption growth from 1890 to 2010.

Table 4 presents evidence suggesting that the relation is strong over this long historical record. First, note that all the variance terms in the first two columns for each measure are again negative, and everything other than raw consumption is significant at the 10% level. The relation with consumption is weaker in terms of adjusted R2 and overall impact, though it is improved slightly after winsorizing. Examining the third column, the coefficient for variance is less negative prior to 1945. The final row shows the p-value for the test that the two variance coefficients are the same and indicates that the post-1945 coefficient is statistically different. This is unsurprising given Romer’s critique that Kendrick-Kuznets’s data artificially increase the size of its fluctuations, making the consumption results difficult to interpret what is reflecting economic fundamentals and what is reflecting data errors.

Table 4

Regressions of the real interest rate on forecasts of growth and uncertainty using historic data

 Consumption 1890-2010 GDP 1871 - 2010 Industrial production 1895 - 2010 
 No correction Winsorised 90% No correction Winsorised 90% No correction Winsorised 90% 
Et[gt+10.098 0.367  0.100 0.560  0.648 0.938  
 (0.18) (0.49)  (0.28) (1.26)  (2.19) (2.23)  
Vart[gt+1−6.150 −7.813  −6.262 −8.734  −1.487 −1.697  
 (−1.53) (−1.68)  (−2.20) (−2.41)  (−2.47) (−2.62)  
Et[gt+1]*(Year≥1945)   0.642   0.188   0.397 
   (0.60)   (0.47)   (0.64) 
Et[gt+1]*(Year<1945)   0.411   0.281   0.840 
   (0.57)   (0.66)   (1.71) 
Vart[gt+1]*(Year≥1945)   −43.083   −15.546   −3.275 
   (−6.42)   (−8.33)   (−6.58) 
Vart[gt+1]*(Year<1945)   −10.297   −6.776   −1.825 
   (−1.19)   (−1.50)   (−1.97) 
Constant 0.026 0.022 0.031 0.034 0.026 0.036 0.024 0.021 0.031 
 (1.86) (1.16) (1.27) (3.58) (2.24) (3.22) (3.20) (2.11) (2.72) 
 
Observations 121 121 121 140 140 140 126 126 126 
Adjusted R2 0.02 0.03 0.18 0.11 0.13 0.20 0.15 0.16 0.25 
R2 excluding Vart[gt+1−0.01 0.00  −0.01 0.00  0.01 0.01  
Impact of 1 SD Vart[gt+1−0.005 −0.006  −0.011 −0.012  −0.011 −0.012  
p-Value Vart: (Y ≥ 1945) = (Y < 1945)   0.001   0.061   0.144 
 Consumption 1890-2010 GDP 1871 - 2010 Industrial production 1895 - 2010 
 No correction Winsorised 90% No correction Winsorised 90% No correction Winsorised 90% 
Et[gt+10.098 0.367  0.100 0.560  0.648 0.938  
 (0.18) (0.49)  (0.28) (1.26)  (2.19) (2.23)  
Vart[gt+1−6.150 −7.813  −6.262 −8.734  −1.487 −1.697  
 (−1.53) (−1.68)  (−2.20) (−2.41)  (−2.47) (−2.62)  
Et[gt+1]*(Year≥1945)   0.642   0.188   0.397 
   (0.60)   (0.47)   (0.64) 
Et[gt+1]*(Year<1945)   0.411   0.281   0.840 
   (0.57)   (0.66)   (1.71) 
Vart[gt+1]*(Year≥1945)   −43.083   −15.546   −3.275 
   (−6.42)   (−8.33)   (−6.58) 
Vart[gt+1]*(Year<1945)   −10.297   −6.776   −1.825 
   (−1.19)   (−1.50)   (−1.97) 
Constant 0.026 0.022 0.031 0.034 0.026 0.036 0.024 0.021 0.031 
 (1.86) (1.16) (1.27) (3.58) (2.24) (3.22) (3.20) (2.11) (2.72) 
 
Observations 121 121 121 140 140 140 126 126 126 
Adjusted R2 0.02 0.03 0.18 0.11 0.13 0.20 0.15 0.16 0.25 
R2 excluding Vart[gt+1−0.01 0.00  −0.01 0.00  0.01 0.01  
Impact of 1 SD Vart[gt+1−0.005 −0.006  −0.011 −0.012  −0.011 −0.012  
p-Value Vart: (Y ≥ 1945) = (Y < 1945)   0.001   0.061   0.144 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1. Growth is measured by consumption (1890–2010), GDP (1871–2010), and industrial production (1895–2010). “No correction” columns use the raw measure, and the column labeled “Winsorized 90%” is winsorized at the 90th percentile. The third column of each measure allows for different coefficients on pre- and post-1945 with the p-value of the test of Vart[gt+1]*(Year≥1945) = Vart[gt+1]*(Year<1945) in the last row. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Using the Romer and Miron and Romer series, which lack the known issues of the consumption data, the pattern with GDP and industrial production remains robust. The adjusted R2 increases when uncertainty is included in the model. The R2 is lower than it was during the shorter period. This is unsurprising, as the regressions cover 140 years, including many different economic episodes, periods, and policies. However, the general pattern remains significant. The growth forecast term is positive in all specifications, but it is only significant at the 5% level when using industrial production. The impact of a one-standard-deviation increase in uncertainty is associated with a greater than 0.01 decrease in the interest rate for both GDP and industrial production. Examining the third column, the coefficients on uncertainty pre-1945 are less negative for both measures, but neither is statistically significantly different at the 5% level from the post-1945 measures.

3.4 Quarterly data

The previous analysis uses annual data as it is available for the longest time period and does not suffer from being seasonally adjusted. As a further robustness check, Table 5 presents the same specification as Table 2, panel B, but using quarterly data. The benefit of the quarterly data is that it is higher frequency and may not suffer from the structural breaks in the data that occur over the long time series (Chapman 1998; Otrok, Ravikumar, and Whiteman 2002). The quarterly data begin in 1947, so major events effecting uncertainty, such as World War II and the Great Depression, are not included. Further concerns about persistent regressors are greater for these regressions as the sampling occurs quarterly as opposed to annually. Finally, the seasonal adjustment may be taking out relevant variation from the data and biasing results (Ferson and Harvey 1992). All growth rates are annualized for comparability with the previous estimates.

Table 5

Quarterly regressions of the real interest rate on forecasts of growth and uncertainty

 Consumption GDP Industrial production 
Et[gt+10.086 0.062 0.019 
 (0.93) (0.70) (0.46) 
Vart[gt+1−64.377 −29.851 −14.882 
 (−3.30) (−1.56) (−5.20) 
Constant 0.011 0.013 0.021 
 (2.15) (1.96) (6.36) 
 
Observations 256 256 256 
Adjusted R2 0.08 0.02 0.26 
R2 excluding Vart[gt+10.01 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.008 −0.005 −0.015 
 Consumption GDP Industrial production 
Et[gt+10.086 0.062 0.019 
 (0.93) (0.70) (0.46) 
Vart[gt+1−64.377 −29.851 −14.882 
 (−3.30) (−1.56) (−5.20) 
Constant 0.011 0.013 0.021 
 (2.15) (1.96) (6.36) 
 
Observations 256 256 256 
Adjusted R2 0.08 0.02 0.26 
R2 excluding Vart[gt+10.01 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.008 −0.005 −0.015 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1947 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The growth rate is annual and the interest rate is annualized. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using four lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Table 5 shows that in general the results remain robust to using the shorter time period in the quarterly sample, though they are weaker for GDP. All three measures of variance are negative, though GDP is no longer significant. Further, in all three specifications, the adjusted R2 increases versus the specification excluding variance.

3.5 Different interest rates

The analysis to this point has utilized the three-month interest rate. Next, the model is estimated using a variety of alternative interest rates. The interest rates with data available for this period are the 30-day interest rate, the three-month interest rate, the six-month interest rate, and the one-year interest rate. Table 7 presents the estimates. Notice that the main patterns discussed in the previous section are all present for each maturity. The expected growth is insignificant, while the variance is negative and significant. The portion of the data explained using the variance versus not, is significantly higher as measured by the adjusted R2. Thus, the choice of short-term interest rate does not appear to be an important factor for the results.

3.6 International data

Finally, I analyze the relation by examining countries other than the United States. Extending the analysis to other countries allows for more observations and alternate time series to analyze whether the U.S. relation is simply an anomaly. Eight different developed countries are examined and estimates for growth and uncertainty are obtained by estimating a separate ARMA-GARCH model for each country as specified by Equation (3).

Table 6 examines each country individually using consumption as a measure of growth in panel A and GDP in panel B. The results are robust and similar to the U.S. results. All eight of the countries have negative coefficients on the uncertainty term, and six of them are significant at the 5% level. When examining GDP in panel B, seven of the eight coefficients on uncertainty have a negative sign and five of the eight are significant at the 5% level.

Table 6

Regressions of the real interest rate on forecasts of growth and uncertainty for countries outside the United States

A. Consumption 
 Belgium Canada France Germany Italy Japan Netherlands United Kingdom 
Et[gt+10.780 0.045 −0.359 0.394 1.450 0.870 −0.063 1.080 
 (1.40) (0.06) (−0.91) (1.64) (2.40) (3.91) (−2.94) (3.98) 
Vart[gt+1−0.807 −19.683 −6.506 −125.537 −31.369 −35.510 −0.345 0.000 
 (−0.03) (−3.91) (−4.21) (−2.07) (−3.77) (−3.72) (−11.64) (−1.85) 
Constant 0.014 0.019 0.026 0.033 −0.033 −0.003 0.006 −0.006 
 (0.99) (0.97) (2.07) (3.30) (−1.47) (−0.45) (0.96) (−0.88) 
 
Observations 61 75 49 56 69 49 68 109 
Adjusted R2 0.00 0.14 0.08 0.12 0.31 0.40 0.27 0.16 
R2 excluding Vart[gt+10.01 −0.01 0.03 −0.02 0.02 0.03 0.05 0.17 
Impact of 1 SD Vart[gt+10.000 −0.013 −0.008 −0.007 −0.083 −0.018 −0.015 −0.001 
 
B. GDP 
 
 Belgium Canada France Germany Italy Japan Netherlands United Kingdom 
 
Et[gt+11.090 −0.126 −0.344 0.431 2.743 0.554 −0.002 −0.150 
 (4.66) (−0.50) (−1.13) (1.56) (3.02) (4.69) (−0.11) (−0.34) 
Vart[gt+110.523 −15.940 −3.546 −2.723 −5.844 −9.335 −0.323 −9.024 
 (0.61) (−3.63) (−2.55) (−1.07) (−4.52) (−3.47) (−5.54) (−0.42) 
Constant −0.004 0.025 0.024 0.007 −0.101 −0.003 0.005 0.020 
 (−0.55) (2.49) (2.33) (1.11) (−3.01) (−0.65) (0.81) (1.01) 
 
Observations 61 75 49 56 69 49 68 109 
Adjusted R2 0.17 0.09 0.07 0.00 0.41 0.40 0.19 −0.01 
R2 excluding Vart[gt+10.18 0.02 0.04 0.00 0.18 0.09 −0.01 −0.01 
Impact of 1 SD Vart[gt+10.001 −0.011 −0.007 −0.002 −0.077 −0.015 −0.015 −0.003 
A. Consumption 
 Belgium Canada France Germany Italy Japan Netherlands United Kingdom 
Et[gt+10.780 0.045 −0.359 0.394 1.450 0.870 −0.063 1.080 
 (1.40) (0.06) (−0.91) (1.64) (2.40) (3.91) (−2.94) (3.98) 
Vart[gt+1−0.807 −19.683 −6.506 −125.537 −31.369 −35.510 −0.345 0.000 
 (−0.03) (−3.91) (−4.21) (−2.07) (−3.77) (−3.72) (−11.64) (−1.85) 
Constant 0.014 0.019 0.026 0.033 −0.033 −0.003 0.006 −0.006 
 (0.99) (0.97) (2.07) (3.30) (−1.47) (−0.45) (0.96) (−0.88) 
 
Observations 61 75 49 56 69 49 68 109 
Adjusted R2 0.00 0.14 0.08 0.12 0.31 0.40 0.27 0.16 
R2 excluding Vart[gt+10.01 −0.01 0.03 −0.02 0.02 0.03 0.05 0.17 
Impact of 1 SD Vart[gt+10.000 −0.013 −0.008 −0.007 −0.083 −0.018 −0.015 −0.001 
 
B. GDP 
 
 Belgium Canada France Germany Italy Japan Netherlands United Kingdom 
 
Et[gt+11.090 −0.126 −0.344 0.431 2.743 0.554 −0.002 −0.150 
 (4.66) (−0.50) (−1.13) (1.56) (3.02) (4.69) (−0.11) (−0.34) 
Vart[gt+110.523 −15.940 −3.546 −2.723 −5.844 −9.335 −0.323 −9.024 
 (0.61) (−3.63) (−2.55) (−1.07) (−4.52) (−3.47) (−5.54) (−0.42) 
Constant −0.004 0.025 0.024 0.007 −0.101 −0.003 0.005 0.020 
 (−0.55) (2.49) (2.33) (1.11) (−3.01) (−0.65) (0.81) (1.01) 
 
Observations 61 75 49 56 69 49 68 109 
Adjusted R2 0.17 0.09 0.07 0.00 0.41 0.40 0.19 −0.01 
R2 excluding Vart[gt+10.18 0.02 0.04 0.00 0.18 0.09 −0.01 −0.01 
Impact of 1 SD Vart[gt+10.001 −0.011 −0.007 −0.002 −0.077 −0.015 −0.015 −0.003 

This table presents regressions for each country of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1. Growth is measured by consumption and GDP. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model for each country. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Table 7

Regressions of various interest rates on forecasts of growth and uncertainty

A. Consumption 
 30 day 3 month 6 month 1 year 
Et[gt+1−1.019 −0.834 0.720 −0.519 
 (−1.12) (−0.88) (1.05) (−0.30) 
Vart[gt+1−58.880 −60.389 −66.215 −74.988 
 (−3.08) (−3.30) (−6.64) (−2.35) 
Constant 0.040 0.038 0.014 0.043 
 (1.89) (1.72) (0.87) (1.08) 
 
Observations 76 76 76 64 
Adjusted R2 0.14 0.14 0.19 0.12 
R2 excluding Vart[gt+10.00 0.00 0.02 0.00 
Impact of 1 SD Vart[gt+1−0.012 −0.012 −0.013 −0.015 
 
B. GDP 
 
 30 day 3 month 6 month 1 year 
 
Et[gt+10.194 0.239 0.213 0.347 
 (0.93) (1.14) (0.78) (0.77) 
Vart[gt+1−12.799 −12.902 −10.576 −14.331 
 (−10.61) (−11.22) (−5.73) (−5.18) 
Constant 0.019 0.020 0.025 0.025 
 (3.14) (3.03) (3.45) (2.00) 
 
Observations 76 76 76 64 
Adjusted R2 0.56 0.55 0.36 0.45 
R2 excluding Vart[gt+1−0.01 −0.01 −0.01 0.15 
Impact of 1 SD Vart[gt+1−0.023 −0.023 −0.019 −0.026 
 
C. Industrial production 
 
 30 day 3 month 6 month 1 year 
 
Et[gt+10.021 0.049 0.305 0.333 
 (0.09) (0.20) (1.70) (1.46) 
Vart[gt+1−2.419 −2.441 −2.360 −6.018 
 (−3.01) (−3.04) (−3.82) (−6.59) 
Constant 0.019 0.020 0.024 0.034 
 (3.74) (3.72) (3.86) (4.29) 
 
Observations 76 76 76 64 
Adjusted R2 0.36 0.35 0.31 0.45 
R2 excluding Vart[gt+10.00 0.00 −0.01 −0.01 
Impact of 1 SD Vart[gt+1−0.018 −0.018 −0.018 −0.045 
A. Consumption 
 30 day 3 month 6 month 1 year 
Et[gt+1−1.019 −0.834 0.720 −0.519 
 (−1.12) (−0.88) (1.05) (−0.30) 
Vart[gt+1−58.880 −60.389 −66.215 −74.988 
 (−3.08) (−3.30) (−6.64) (−2.35) 
Constant 0.040 0.038 0.014 0.043 
 (1.89) (1.72) (0.87) (1.08) 
 
Observations 76 76 76 64 
Adjusted R2 0.14 0.14 0.19 0.12 
R2 excluding Vart[gt+10.00 0.00 0.02 0.00 
Impact of 1 SD Vart[gt+1−0.012 −0.012 −0.013 −0.015 
 
B. GDP 
 
 30 day 3 month 6 month 1 year 
 
Et[gt+10.194 0.239 0.213 0.347 
 (0.93) (1.14) (0.78) (0.77) 
Vart[gt+1−12.799 −12.902 −10.576 −14.331 
 (−10.61) (−11.22) (−5.73) (−5.18) 
Constant 0.019 0.020 0.025 0.025 
 (3.14) (3.03) (3.45) (2.00) 
 
Observations 76 76 76 64 
Adjusted R2 0.56 0.55 0.36 0.45 
R2 excluding Vart[gt+1−0.01 −0.01 −0.01 0.15 
Impact of 1 SD Vart[gt+1−0.023 −0.023 −0.019 −0.026 
 
C. Industrial production 
 
 30 day 3 month 6 month 1 year 
 
Et[gt+10.021 0.049 0.305 0.333 
 (0.09) (0.20) (1.70) (1.46) 
Vart[gt+1−2.419 −2.441 −2.360 −6.018 
 (−3.01) (−3.04) (−3.82) (−6.59) 
Constant 0.019 0.020 0.024 0.034 
 (3.74) (3.72) (3.86) (4.29) 
 
Observations 76 76 76 64 
Adjusted R2 0.36 0.35 0.31 0.45 
R2 excluding Vart[gt+10.00 0.00 −0.01 −0.01 
Impact of 1 SD Vart[gt+1−0.018 −0.018 −0.018 −0.045 

This table presents regressions of various interest rates in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively, using data from 1934 until 2010. The interest rate used is indicated by the column header. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top number is the coefficient, and the lower number in parentheses is the t-statistic.

The international results also offer some support for the link between growth and the interest rate, albeit not as robustly as that between the interest rate and uncertainty. Examining the results utilizing consumption in panel A, I find that six of the eight coefficients on growth are positive and three are significant at the 5% level. Examining the results using GDP in panel B, I find that four of the eight are positive with three significant at the 5% level.

4. Using Alternative Forecasts

4.1 Using ex post growth

The growth estimates to this point are from time-series models that may not accurately reflect the expectations of the market. The base interest rate relation implies that growth should be positively correlated with the interest rate, but none of the β1 coefficients are significant, and, in some cases, the point estimates are negative. Both the growth and uncertainty term are measured with error, and this should attenuate the estimates if they are uncorrelated and random. It is possible that the forecast of growth is measured with more error than the variance term and that the significance of the variance term is driven by unmeasured variation in growth.

To test for this, Table 8, panel A, presents estimates of Equation (1) using ex post realized growth instead of forecasts. This introduces a different bias in the forecast term as the agent does not know this value at time t, but if the results are being driven by systematically failing to measure the agent’s expectations, the ex post realized value should present a worst-case scenario for the errors in variables bias. All of the previous results remain materially unchanged when using ex post growth. Thus, it is unlikely that measurement error of the forecast term is driving the uncertainty coefficient.

Table 8

Regressions of the real interest rate on uncertainty, ex post growth, and expert forecasts of growth

A. Ex-post growth 
 Consumption GDP Industrial production 
Et[gt+10.105 0.057 0.041 
 (0.36) (1.17) (1.16) 
Vart[gt+1−56.658 −12.853 −2.572 
 (−5.04) (−8.68) (−4.16) 
Constant 0.017 0.023 0.021 
 (1.91) (4.95) (3.90) 
 
Observations 76 76 76 
Adjusted R2 0.12 0.54 0.35 
R2 excluding Vart[gt+10.00 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.011 −0.023 −0.019 
 
B. Expert forecasts 
 
 Consumption GDP Industrial production 
 
Ei,t[gt+10.144 −0.084 −0.002 
 (0.91) (−0.70) (−0.03) 
Vart[gt+1−56.619 −5.672 −5.446 
 (−5.74) (−0.52) (−4.30) 
Constant 0.025 0.022 0.030 
 (4.78) (3.44) (6.76) 
 
Observations 2,955 4,856 4,710 
Number of years 29 42 42 
Adjusted R2 0.19 0.01 0.11 
R2 excluding Vart[gt+10.03 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.011 −0.010 −0.041 
 
Panel C. Rolling forecasts 
 
 Consumption GDP Industrial production 
 
Et[gt+10.874 −0.007 0.082 
 (1.80) (−0.02) (0.54) 
Vart[gt+1−2.677 −0.438 −0.854 
 (−4.26) (−0.48) (−5.14) 
Constant −0.003 0.016 0.016 
 (−0.27) (1.70) (2.18) 
 
Observations 60 60 60 
Adjusted R2 0.16 −0.03 0.10 
R2 excluding Vart[gt+1−0.01 −0.02 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.001 −0.008 
A. Ex-post growth 
 Consumption GDP Industrial production 
Et[gt+10.105 0.057 0.041 
 (0.36) (1.17) (1.16) 
Vart[gt+1−56.658 −12.853 −2.572 
 (−5.04) (−8.68) (−4.16) 
Constant 0.017 0.023 0.021 
 (1.91) (4.95) (3.90) 
 
Observations 76 76 76 
Adjusted R2 0.12 0.54 0.35 
R2 excluding Vart[gt+10.00 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.011 −0.023 −0.019 
 
B. Expert forecasts 
 
 Consumption GDP Industrial production 
 
Ei,t[gt+10.144 −0.084 −0.002 
 (0.91) (−0.70) (−0.03) 
Vart[gt+1−56.619 −5.672 −5.446 
 (−5.74) (−0.52) (−4.30) 
Constant 0.025 0.022 0.030 
 (4.78) (3.44) (6.76) 
 
Observations 2,955 4,856 4,710 
Number of years 29 42 42 
Adjusted R2 0.19 0.01 0.11 
R2 excluding Vart[gt+10.03 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.011 −0.010 −0.041 
 
Panel C. Rolling forecasts 
 
 Consumption GDP Industrial production 
 
Et[gt+10.874 −0.007 0.082 
 (1.80) (−0.02) (0.54) 
Vart[gt+1−2.677 −0.438 −0.854 
 (−4.26) (−0.48) (−5.14) 
Constant −0.003 0.016 0.016 
 (−0.27) (1.70) (2.18) 
 
Observations 60 60 60 
Adjusted R2 0.16 −0.03 0.10 
R2 excluding Vart[gt+1−0.01 −0.02 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.001 −0.008 

Panel A contains regressions as described in Table 2, but the realized ex post value for growth is used. Panel B presents panel regressions by year and individual expert forecast. The log of one, plus the nominal three-month Treasury-bill rate in year t, minus the expert forecast of inflation, is regressed on forecasts of the log of one, plus growth and the variance of the growth in year t+1 (calculated from an ARMA(1,1)-GARCH(1,1) model) using data from 1982 until 2010 for consumption and 1969–2010 for GDP and industrial production. Standard errors are clustered by year and by expert ID. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

4.2 Using professional forecasts

Table 8, panel B, presents another robustness check to the growth forecast by using an alternative measure of growth from the Survey of Professional Forecasters. This table estimates the panel regression:  

(4)
ri,t=β0+β1Ei,tgt+1+β2Vartgt+1,
where i indexes each individual expert and t indexes time. ri,t is an expert’s forecast of the real risk-free rate based on their estimate of inflation, and Ei,tgt+1 is their estimate of growth. By using a panel format, it is possible to match a given expert’s estimate of the risk-free rate with their estimate of growth.9 The variance term does not vary by expert and is specified by Equation (3). Standard errors are clustered by both time and expert. Using the same measure of uncertainty with different measures of inflation and growth serves as another test for whether systematically biased estimates from the ARMA model for growth are responsible for the negative relation between the interest rate and uncertainty.

The results using the expert forecast data are consistent with those presented previously. While GDP is no longer statistically significant, the β2 coefficients for consumption and industrial production are strong with a t-statistic greater than four. All three are still economically significant, with a one-standard-deviation increase in uncertainty being associated with decreases in the risk-free rate of 0.011, 0.010, and 0.041 for consumption, GDP, and industrial production, respectively. Even though there are fewer years, the main results are still present.

Aside from weak forecasts, another concern is that the time-series model induces a look-ahead bias that accounts for the results. The baseline model is estimated using data through the entire sample. If the agent knows this model at time t, the agent can make forecasts using it, but perhaps the model is not known and the results are biased.

Figure 4 graphs the forecasts from the time-series model versus the median expert forecast for the survey of professional forecasters. The expert forecasts were made by professionals using only ex ante knowledge. While not identical, the two series move closely together throughout the period in which there are data for both. This suggests that the forecasts from the model are not purely driven by a look-ahead bias, at least for these years.

Figure 4

Expert and time-series forecasts

This figure plots forecasts of the log of one, plus expected economic growth from time-series models and from expert forecasts. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. The time-series model line shows forecasts from an ARMA(1,1)-GARCH(1,1). The expert line plots the median forecast of growth from the Survey of Professional Forecasters.

Figure 4

Expert and time-series forecasts

This figure plots forecasts of the log of one, plus expected economic growth from time-series models and from expert forecasts. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. The time-series model line shows forecasts from an ARMA(1,1)-GARCH(1,1). The expert line plots the median forecast of growth from the Survey of Professional Forecasters.

As a further test, Table 8, panel C, estimates the relation in which the forecasting model is fit using data only for the years up to time t to forecast period t+1. The model is estimated separately for each year, adding in the new year of data and starting with a minimum of fifteen observations. Using this rolling estimation technique, negative and significant coefficients are found for consumption and industrial production, and GDP is no longer significant. While slightly weaker, this suggests that a look-ahead bias is not responsible for the empirical relation.

4.3 Using the VIX

This section examines the robustness of the previous results by using ex ante forecasts of both growth and uncertainty. The VIX index is utilized to proxy for uncertainty. The VIX corresponds to stock market volatility, which, while related to economic uncertainty, is not a perfect proxy. That being said, the VIX is probably the best ex ante forecaster of economic uncertainty at our disposal (e.g., Bloom 2009). The major shortcoming for this paper is that the data on the VIX begin in 1990. Thus, there is only a short time series to estimate the relation, so VIX is not used as the baseline measure of uncertainty, even though it would be an ideal measure if it were available for the full sample period.

To deal with the short time series of data, monthly forecasts from the Leading Index for the United States and the University of Michigan Consumer Sentiment Index are used for growth. This section, and this section only, utilizes nominal values for all variables. The VIX measure is nominal volatility, and the horizon of consumer sentiment is unclear. Given the stable low level of inflation over this period, similar results in both significance and magnitude are obtained assuming a number of different specifications for inflation.

The VIX, the leading index, and the consumer sentiment index are each constructed using data available at the time they are reported and are used in practice for forecasting purposes. Thus, finding an inverse relation between the VIX and the interest rate underscores that the previous analysis is not driven by the assumption of a specific time-series structure.

Table 9 presents estimates for the relation utilizing the VIX. Panel A utilizes a Newey-West regression framework, and panel B utilizes the mARM correction (discussed in Section 7) to deal with persistent regressors. In all specifications the relation between the interest rate and the VIX is negative and significant. The relation between the Leading Index and the interest rate is not significant, while the consumer sentiment index is positive and significant. Thus, the interest rate and uncertainty retain a negative and significant relation, even when examining a shorter period of time, utilizing different data, and utilizing different techniques.

Table 9

VIX regressions of the real interest rate on forecasts of growth and uncertainty

A. Newey regression 
 Leading index for the United States Consumer sentiment 
Et[gt+10.024 0.012 
 (0.74) (3.71) 
VIX −0.014 −0.011 
 (−2.18) (−2.81) 
Constant 0.012 0.013 
 (6.35) (13.46) 
 
Observations 246 246 
Adjusted R2 0.08 0.21 
R2 excluding VIX 0.09 0.19 
Impact of 1 SD VIX −0.001 −0.001 
 
B. mARM procedure 
 
 Leading index for the United States Consumer sentiment 
 
Et[gt+10.026 0.013 
 (1.17) (6.46) 
VIX −0.018 −0.014 
 (−2.82) (−3.30) 
Residual Et[gt+1−0.166 −0.012 
 (−2.64) (−2.06) 
Residual Vart[gt+10.012 0.012 
 (1.30) (1.53) 
Constant 0.012 0.014 
 (7.64) (16.11) 
 
Observations 245 245 
Adjusted R2 0.11 0.23 
Impact of 1 SD VIX −0.001 −0.001 
A. Newey regression 
 Leading index for the United States Consumer sentiment 
Et[gt+10.024 0.012 
 (0.74) (3.71) 
VIX −0.014 −0.011 
 (−2.18) (−2.81) 
Constant 0.012 0.013 
 (6.35) (13.46) 
 
Observations 246 246 
Adjusted R2 0.08 0.21 
R2 excluding VIX 0.09 0.19 
Impact of 1 SD VIX −0.001 −0.001 
 
B. mARM procedure 
 
 Leading index for the United States Consumer sentiment 
 
Et[gt+10.026 0.013 
 (1.17) (6.46) 
VIX −0.018 −0.014 
 (−2.82) (−3.30) 
Residual Et[gt+1−0.166 −0.012 
 (−2.64) (−2.06) 
Residual Vart[gt+10.012 0.012 
 (1.30) (1.53) 
Constant 0.012 0.014 
 (7.64) (16.11) 
 
Observations 245 245 
Adjusted R2 0.11 0.23 
Impact of 1 SD VIX −0.001 −0.001 

This table presents regressions of the three-month Treasury-bill rate in year t on growth forecasts measured by the variable indicated in the column header and variance measured by the VIX index using data from 1990 until 2010. Growth Measures are in logs. In panel A Newey-West regressions are run with 3 lags. In panel B the mARM persistence correction is used. Standard errors in panel B for the coefficients on Et[gt+1] and Vart[gt+1] are mARM corrected and all others are White robust errors. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

4.4 Using stock market volatility

The VIX is constructed to be an ex ante forecaster of stock market volatility over the next 30 days, but it is only available for a limited period of time. To extend the period of analysis I examine various measures of stock market volatility, calculated as the variance of daily returns in the CRSP value-weighted index. Industrial production is used as the measure of growth as it is available monthly.10 Three measures of market volatility are examined. In the first column of Table 10, realized volatility over the prior period is used (the previous year in panel A and the previous month in panel B). In the second column I use the predicted market volatility in the next year in panel A and the next month in panel B from an ARMA(1,1) model. At the monthly frequency, this is perhaps the closest analog to the VIX as it is a prediction of future volatility over the next month. In the final column, I use the realized volatility over the next year in panel A and the next month in panel B.

Table 10

Market volatility regressions of the real interest rate on forecasts of growth and uncertainty

A. Annual 
 Prior year Predicted next year Actual next year 
Et[gt+1−0.291 −0.273 −0.220 
 (−1.11) (−1.04) (−0.82) 
Vart[gt+1−63.129 −86.229 −10.941 
 (−3.02) (−2.60) (−0.40) 
Constant 0.016 0.018 0.010 
 (2.07) (2.34) (1.26) 
 
Observations 76 76 76 
Adjusted R2 0.02 0.01 −0.02 
R2 excluding Vart[gt+10.00 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.006 −0.005 −0.001 
 
B. Monthly 
 
 Prior month Predicted next month Actual next month 
 
Et[gt+1−0.236 −0.238 −0.243 
 (−2.39) (−2.40) (−2.45) 
Vart[gt+1−4.214 −7.039 −4.316 
 (−2.35) (−2.42) (−2.12) 
Constant 0.010 0.010 0.010 
 (20.67) (19.18) (20.33) 
 
Observations 684 684 684 
Adjusted R2 0.03 0.03 0.03 
R2 excluding Vart[gt+10.02 0.02 0.02 
Impact of 1 SD Vart[gt+1−0.001 −0.001 −0.001 
A. Annual 
 Prior year Predicted next year Actual next year 
Et[gt+1−0.291 −0.273 −0.220 
 (−1.11) (−1.04) (−0.82) 
Vart[gt+1−63.129 −86.229 −10.941 
 (−3.02) (−2.60) (−0.40) 
Constant 0.016 0.018 0.010 
 (2.07) (2.34) (1.26) 
 
Observations 76 76 76 
Adjusted R2 0.02 0.01 −0.02 
R2 excluding Vart[gt+10.00 0.00 0.00 
Impact of 1 SD Vart[gt+1−0.006 −0.005 −0.001 
 
B. Monthly 
 
 Prior month Predicted next month Actual next month 
 
Et[gt+1−0.236 −0.238 −0.243 
 (−2.39) (−2.40) (−2.45) 
Vart[gt+1−4.214 −7.039 −4.316 
 (−2.35) (−2.42) (−2.12) 
Constant 0.010 0.010 0.010 
 (20.67) (19.18) (20.33) 
 
Observations 684 684 684 
Adjusted R2 0.03 0.03 0.03 
R2 excluding Vart[gt+10.02 0.02 0.02 
Impact of 1 SD Vart[gt+1−0.001 −0.001 −0.001 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and measures of the stock market volatility using data from 1934 until 2010. Growth is measured by industrial production, and market volatility is measured as the variance of daily market returns. Panel A examines annual data, and panel B examines monthly. In panel A (B) the first column examines realized variance of the market in the prior year (month), the second column forecasts variance in the next year (month) using an AMRMA(1,1) model, and the third column examines the realized variance over the next year (month). The row labeled “Impact of 1 SD MV” is the standard deviation of the market variance measure, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Table 10 finds a strong inverse relation between the level of the risk-free interest rate and the various measure of stock market volatility. Examining the annual measure in panel A, I find that all three columns have a negative coefficient, though the relation with ex post realized market volatility is smaller and not statistically significant. Examining monthly data in panel B, similar to the 30-day forecast from the VIX, there is a strong inverse relation for all three measures of market volatility. The results suggest that the short-time period of the VIX does not account for the negative relation found in Table 9 and that the real interest rate has a strong inverse relation with a variety of proxies for uncertainty.

5. Inflation Risk

The previous discussion assumes a risk-free rate for which there is no perfect analog in the data. For the initial analysis, inflation is predicted using an ARMA(1,1) process and the real rate is constructed by subtracting predicted inflation from the nominal interest rate. Using this method, the data suggest that economic growth risk is important for understanding the general equilibrium relation between economic uncertainty, economic growth, and the risk-free interest rate. This ignores any inflation risk premium. The simple discrete-time representative agent model with conditionally lognormal disturbances discussed in Appendix A can also be restated using variables for nominal interest rates, Rj,tN, and inflation, It+j, which is unknown at time t as  

(5)
EδjCt+jCt-α1+Rj,tN1+It+j=1.

Rearranging this and taking logs, I obtain:  

(6)
ln1+Rj,tN-Etln1+It+j=-jlnδ+αEtlnCt+jCt-α22VartlnCt+jCt+12Vartln1+It+j+α2CovtlnCt+jCt,ln1+It+j.

With inflation uncertainty, there are new terms for the variance of inflation and the covariance between consumption growth and inflation. To examine the impact of inflation uncertainty on the results, I estimate:  

(7)
rt=β0+β1Etgt+1+β2Vartgt+1+β3Vartit+1+β4Covtgt+1,it+1,
where it+1 is the log of one, plus inflation. The variance of inflation measures the uncertainty of inflation from t to t+1. The covariance captures the real impact of inflation, through its relation to real economic growth.

For this reduced-form empirical model, both inflation and consumption growth are modeled using an ARMA(1,1)-GARCH(1,1) specification as described by Equation (3). The covariance term is modeled by assuming the residual from the forecast of inflation, multiplied by the residual from the forecast of growth, follows an ARMA(1,1) process. Thus, the covariance term is forecast using the equation:  

(8)
Covt+1gt+1,it+1=γ+α1ϵtgϵti+α2σtgi,
where σtgi is the covariance between growth and inflation at time t.

The first column of Table 11, panel A, presents the estimates of Equation (7) using the three-month interest rate and the three measures of growth. The regression includes the variables in Equation (1) and also the variance of the inflation rate and the covariance term. In all estimates of Equation (7), the coefficient on uncertainty is negative and significant at the 10% level. Adding the variance of inflation decreases the adjusted R2 for all three measures of growth as compared to Table 2, panel B. A one-standard-deviation increase in growth is associated with decreases in the interest rate of -0.011, -0.027, and -0.15 for each measure of growth. These are generally consistent with the numbers obtained when the inflation terms are not included. Thus, the results do not appear driven by simplifying assumptions related to changing the nominal rate to real.11

Table 11

Controlling for inflation risk in regression of the real interest rate on forecasts of growth and uncertainty

A. Inflation risk from time-series model 
 Consumption GDP Industrial production 
Et[gt+1−1.492 0.524 0.247 
 (−1.24) (1.44) (0.67) 
Vart[gt+1−53.782 −15.075 −2.005 
 (−1.75) (−4.78) (−1.90) 
Vart[it+1−13.995 5.224 −4.097 
 (−3.04) (0.79) (−0.93) 
Covt[gt+1,it+1132.277 −22.711 −14.038 
 (2.01) (−1.04) (−1.04) 
Constant 0.082 0.009 0.016 
 (2.26) (0.89) (2.22) 
 
Observations 76 76 76 
Adjusted R2 0.10 0.44 0.18 
Impact of 1 SD Vart[gt+1−0.011 −0.027 −0.015 
 
B. Inflation risk as variance of expert inflation forecasts 
 
 Consumption GDP Industrial production 
 
Et[gt+10.222 −0.086 0.004 
 (1.40) (−0.72) (0.07) 
Vart[gt+1−63.389 −4.928 −6.574 
 (−5.22) (−0.40) (−4.18) 
Vart[it+12.350 −8.063 32.892 
 (3.52) (−0.21) (0.92) 
Constant 0.009 0.023 0.029 
 (1.34) (3.70) (6.51) 
 
Observations 2,955 4,856 4,710 
Adjusted R2 0.29 0.01 0.13 
Impact of 1 SD Vart[gt+1−0.013 −0.009 −0.049 
A. Inflation risk from time-series model 
 Consumption GDP Industrial production 
Et[gt+1−1.492 0.524 0.247 
 (−1.24) (1.44) (0.67) 
Vart[gt+1−53.782 −15.075 −2.005 
 (−1.75) (−4.78) (−1.90) 
Vart[it+1−13.995 5.224 −4.097 
 (−3.04) (0.79) (−0.93) 
Covt[gt+1,it+1132.277 −22.711 −14.038 
 (2.01) (−1.04) (−1.04) 
Constant 0.082 0.009 0.016 
 (2.26) (0.89) (2.22) 
 
Observations 76 76 76 
Adjusted R2 0.10 0.44 0.18 
Impact of 1 SD Vart[gt+1−0.011 −0.027 −0.015 
 
B. Inflation risk as variance of expert inflation forecasts 
 
 Consumption GDP Industrial production 
 
Et[gt+10.222 −0.086 0.004 
 (1.40) (−0.72) (0.07) 
Vart[gt+1−63.389 −4.928 −6.574 
 (−5.22) (−0.40) (−4.18) 
Vart[it+12.350 −8.063 32.892 
 (3.52) (−0.21) (0.92) 
Constant 0.009 0.023 0.029 
 (1.34) (3.70) (6.51) 
 
Observations 2,955 4,856 4,710 
Adjusted R2 0.29 0.01 0.13 
Impact of 1 SD Vart[gt+1−0.013 −0.009 −0.049 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1, as well as inflation forecasts using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West to using three lags. The top number is the coefficient, and the lower number in parentheses is the t-statistic.

The estimates of inflation risk in panel A are from a time-series model that may not yield appropriate forecasts. Table 11, panel B, repeats the panel analysis described by Equation (7), where inflation risk is proxied using the variance of expert forecasts of inflation. All coefficients remain negative, though GDP is insignificant and a one-standard-deviation change in uncertainty is associated with economically meaningful changes to the risk-free rate.

6. Econometric Issues

All of the variables in the above regression have a degree of time-series persistence. Indeed, the GARCH model was created to deal with data series that exhibit persistence in volatility (Engle 1982; McNees 1979). Using one-year intervals and Newey-West standard errors, the previous analysis attempts to control for this. This section further examines the issue to see if it is sufficient, and after a number of tests to examine the impact of persistent regressors, all of the results remain.

First, a conservative, back-of-the-envelope result is presented taking an observation once every five years instead of annually. The five-year sampling interval is chosen to be large enough to mitigate concerns about persistent regressors. If the results disappear, it raises the concern that the empirical relation is driven by econometric bias rather than a true economic relation. Further, this test discards a large amount of data to obtain a long sampling interval. Even if the full sample results are not being driven by persistent regressors, it is possible that no relation will be found simply due to the small sample size of sixteen observations. Thus, this is a conservative test that is biased toward rejecting the relation even if persistence is not responsible for the results.

Table 12, panel A, presents estimates taking one observation every five years and shows that the results are largely unchanged. The data start in 1935, examining 1940, 1945, and so on until 2010. Compared to Table 2, the coefficient on uncertainty is more negative on consumption, less negative on GDP, and slightly more negative on industrial production. All coefficients are statistically significant, and all estimates of the impact of a one-standard-deviation change in variance are economically significant. All adjusted R2 are higher when uncertainty is included in the regression. Thus, the back of the envelope test suggests that the results are not being spuriously driven by persistent regressors.

Table 12

Persistence tests for regressions of the real interest rate on forecasts of growth and uncertainty

A. Five-year interval 
 Consumption GDP Industrial production 
Et[gt+10.334 0.807 0.894 
 (0.41) (1.53) (3.12) 
Vart[gt+1−80.626 −11.688 −2.589 
 (−3.04) (−2.73) (−4.58) 
Constant 0.023 0.010 0.013 
 (1.03) (0.96) (2.02) 
 
Observations 16 16 16 
Adjusted R2 0.27 0.31 0.37 
R2 excluding Vart[gt+10.10 0.00 −0.07 
Impact of 1 SD Vart[gt+1−0.016 −0.021 −0.019 
 
B. mARM procedure 
 
 Consumption GDP Industrial production 
 
Et[gt+1−2.491 0.386 0.158 
 (−1.01) (0.80) (0.34) 
Vart[gt+1−140.360 −12.364 −2.897 
 (−3.43) (−6.22) (−5.63) 
Constant 0.092 0.016 0.022 
 (1.68) (1.86) (3.54) 
 
Observations 75 75 75 
Adjusted R2 0.17 0.55 0.38 
Impact of 1 SD Vart[gt+1−0.028 −0.022 −0.022 
A. Five-year interval 
 Consumption GDP Industrial production 
Et[gt+10.334 0.807 0.894 
 (0.41) (1.53) (3.12) 
Vart[gt+1−80.626 −11.688 −2.589 
 (−3.04) (−2.73) (−4.58) 
Constant 0.023 0.010 0.013 
 (1.03) (0.96) (2.02) 
 
Observations 16 16 16 
Adjusted R2 0.27 0.31 0.37 
R2 excluding Vart[gt+10.10 0.00 −0.07 
Impact of 1 SD Vart[gt+1−0.016 −0.021 −0.019 
 
B. mARM procedure 
 
 Consumption GDP Industrial production 
 
Et[gt+1−2.491 0.386 0.158 
 (−1.01) (0.80) (0.34) 
Vart[gt+1−140.360 −12.364 −2.897 
 (−3.43) (−6.22) (−5.63) 
Constant 0.092 0.016 0.022 
 (1.68) (1.86) (3.54) 
 
Observations 75 75 75 
Adjusted R2 0.17 0.55 0.38 
Impact of 1 SD Vart[gt+1−0.028 −0.022 −0.022 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. In panel A one observations is used every five years starting in 1935 through the end of 2010. In panel B the mARM persistence correction is used. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors in panel B for the coefficients on Et[gt+1] and Vart[gt+1] are mARM corrected, and all others are White robust errors. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Kendall (1954) and Stambaugh (1999) show that there is a finite sample bias when regressors are autoregressive with errors that are correlated with the dependent variable. Amihud and Hurvich (2004) and Amihud, Hurvich, and Wang (2009) develop the mARM method for estimating unbiased coefficients and test statistics of multivariate models with persistent regressors. Appendix B contains a description of the implementation used here.

Table 12, panel B, presents the adjusted regression. After the correction, all estimates of the coefficient on uncertainty are negative and significant. The coefficients on consumption, GDP, and industrial production have t-statistics of -4.00, -6.22, and -5.63, respectively. Also, note that the order of magnitude of the estimates is similar, suggesting that bias is not greatly impacting the coefficients.

7. Implications of Specific Models

7.1 External habit

Campbell and Cochrane (1999) examine the impact of changing risk aversion caused by deviations from slow-moving external habit. Adding habit can create additional habit-based components to the interest rate process. This section examines whether these components account for the negative relation found between uncertainty and the interest rate. After controlling for habit, a robust relation between the interest rate and uncertainty remains. After accounting for uncertainty, the habit-based terms add little explanatory power.

The benchmark model of habits (Campbell and Cochrane 1999) cannot directly speak to the analysis in this paper as it assumes i.i.d. consumption growth. Thus, there are no fluctuations in consumption volatility. Prior to specifying the sensitivity function, λst, where st is the log of surplus consumption stlnCt-XtCt, the risk-free interest rate contains additional linear terms to the affine function found in this paper based on habit. In the Campbell and Cochrane (1999) model, the sensitivity function is chosen so that these terms perfectly offset. The constant expectations of growth and uncertainty combined with this sensitivity function yield a constant risk-free rate.

Alternative habit-based models feature risk-free rates that vary with habit, but offer differing predictions in how it does so. For example, Wachter (2006) builds a model of the term structure based on external habit formation utilizing a risk-free rate that is linear in surplus consumption. For the model to generate an upward-sloping yield curve and a positive risk premia on real bonds, the intertemporal smoothing motive must dominate. Wachter offers support for this hypothesis by regressing the interest rate on surplus consumption and finding a negative coefficient. On the other hand, Verdelhan (2010) builds a model with interest rates that are procyclical in order to explain the uncovered interest rate parity puzzle. Thus, the dominant motive has important implications for the cyclicality of the risk-free rate and there is disagreement on what it should be.

I replicate the Wachter result using annual data and the longer time period studied in this paper to show that without further controls, the negative relation between surplus consumption and the interest rate is robust. I use the same proxy for surplus consumption, a slow-moving weighted average of past consumption, and estimate  

(9)
rt=β0+β1j=110φjgt-j+ϵt+1.

Surplus consumption is measured over a ten-year period as j=110φjgt-j, where φ = (0.97)4. Table 13, panel A, presents the results. The relation is negative and is significant for GDP and industrial production. Thus, both economic uncertainty and surplus consumption are negatively correlated with the level of the interest rate.

Table 13

Controlling for surplus consumption in regressions of the real interest rate on forecasts of growth and uncertainty

A. Regressions with only surplus consumption 
 Consumption GDP Industrial production 
Surplus consumption −0.151 −0.101 −0.053 
 (−1.12) (−4.35) (−2.66) 
Constant 0.036 0.029 0.017 
 (1.44) (4.02) (2.36) 
 
Observations 71 71 76 
Adjusted R2 0.04 0.21 0.12 
 
B. Regressions with surplus consumption, growth, and uncertainty 
 
 Consumption GDP Industrial production 
 
Et[gt+1 −1.236  0.339  0.066 
  (−0.80)  (1.77)  (0.30) 
Vart[gt+1−81.894 −100.863 −13.555 −13.626 −2.185 −2.210 
 (−4.50) (−3.25) (-6.92) (−9.10) (−2.89) (−2.75) 
Surplus consumption −0.136 −0.094 0.020 0.010 −0.023 −0.023 
 (−1.23) (−0.93) (0.81) (0.47) (−1.03) (−1.05) 
Constant 0.048 0.069 0.022 0.017 0.024 0.023 
 (2.37) (1.91) (3.30) (2.24) (4.03) (3.53) 
 
Observations 71 71 71 71 76 76 
Adjusted R2 0.16 0.16 0.57 0.57 0.36 0.36 
Impact of 1 SD Vart[gt+1−0.011 −0.013 -0.024 −0.024 −0.016 −0.017 
A. Regressions with only surplus consumption 
 Consumption GDP Industrial production 
Surplus consumption −0.151 −0.101 −0.053 
 (−1.12) (−4.35) (−2.66) 
Constant 0.036 0.029 0.017 
 (1.44) (4.02) (2.36) 
 
Observations 71 71 76 
Adjusted R2 0.04 0.21 0.12 
 
B. Regressions with surplus consumption, growth, and uncertainty 
 
 Consumption GDP Industrial production 
 
Et[gt+1 −1.236  0.339  0.066 
  (−0.80)  (1.77)  (0.30) 
Vart[gt+1−81.894 −100.863 −13.555 −13.626 −2.185 −2.210 
 (−4.50) (−3.25) (-6.92) (−9.10) (−2.89) (−2.75) 
Surplus consumption −0.136 −0.094 0.020 0.010 −0.023 −0.023 
 (−1.23) (−0.93) (0.81) (0.47) (−1.03) (−1.05) 
Constant 0.048 0.069 0.022 0.017 0.024 0.023 
 (2.37) (1.91) (3.30) (2.24) (4.03) (3.53) 
 
Observations 71 71 71 71 76 76 
Adjusted R2 0.16 0.16 0.57 0.57 0.36 0.36 
Impact of 1 SD Vart[gt+1−0.011 −0.013 -0.024 −0.024 −0.016 −0.017 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1, as well as surplus consumption using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. Panel A presents regressions of the interest rate on surplus consumption for various measures of growth. Panel B adds forecasts of growth and uncertainty. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top number is the coefficient, and the lower number in parentheses is the t-statistic.

While the i.i.d. consumption growth in the above models of habit cannot directly speak to the analysis in this paper, a handful of papers have relaxed this assumption (e.g., Bekaert 1996; Bekaert and Grenadier 2001; Bekaert and Engstrom 2009; Ermolov 2014). For a recent example, Ermolov (2014) builds a habit-based model with heteroscedastic consumption growth shocks that yields a risk-free rate of the form:  

(10)
rt=β0+β1st+β2Vartgt+1
after collapsing a number of constant parameters into the β terms. Consistent with the other models discussed in this paper, the model predicts β2<0. Thus, in this model, the investor wants to smooth consumption based on their habit st, but exhibits precautionary savings with respect to uncertainty.

In the empirical analysis it could be that uncertainty captures the impact of surplus consumption or surplus consumption captures the impact of uncertainty, so it is necessary to control for both to understand the relation these variables have with the interest rate. First, I estimate the regression suggested by the Ermolov model, which takes the Wachter regression from Equation (10) and adds uncertainty to estimate  

(11)
rt=β0+β1j=110φjgt-j+β2Vartgt+1+ϵt+1.

Second, I also allow for time-varying growth and estimate  

(12)
rt=β0+β1Etgt+1+β2Vartgt+1+β3j=110φjgt-j.

Table 13, panel B, shows a strong relation between uncertainty and the risk-free rate and a weak relation between the risk-free rate and habit after both are included. The first column under each growth measure presents estimates utilizing Equation (11). After controlling for surplus consumption, all three measures of growth yield negative and significant coefficients on uncertainty that are a similar magnitude to those found without controlling for surplus consumption. Compared to the specification with surplus consumption alone, adding uncertainty and growth increases the adjusted R2 from 0.04 to 0.16 using consumption, 0.21 to 0.57 using GDP, and 0.12 to 0.36 using industrial production. The next column allows for time-varying growth and shows a similar pattern. The coefficients on surplus consumption move from negative and significant in panel A to roughly zero in all three specifications in panel B. The signs of the point estimates on surplus consumption are not consistent across growth measures, and none are significant.

Surplus consumption is unable to account for the negative relation between uncertainty and the interest rate. After uncertainty is added to the regressions, the relation between surplus consumption and the interest rate is weak. This suggests that changes to risk aversion based on habit are not a major factor for understanding the interest rate, while uncertainty and the interest rate have a strong connection.

7.2 Long-run risks

The long-run risks model is one of the few models to include time-varying uncertainty. The interest rate is modeled as linear in growth and uncertainty, so the results in this paper add to the empirical literature on long-run risks (Bansal and Yaron 2004; Bansal, Khatchatrian, and Yaron 2005; Bansal, Kiku, and Yaron 2007, 2012; Ferson, Nallareddy, and Xie 2013; Jagannathan and Marakani 2011).

The long-run risks interest rate is linear in growth and uncertainty and, as discussed in Appendix A, can be written as  

(13)
rt=Af+A1,fEtgt+1+A2,fVartgt+1.

Each of these As is a constant parameter of the model. While the coefficients include model parameters different from the motivating example in Section 2.1, the interest rate is the linear function of growth and uncertainty examined in this paper.

In the long-run risks model, asset prices are impacted by a persistent predictable component of consumption growth and a persistent predictable component of uncertainty. Thus, similar to Beeler and Campbell (2012), this paper finds that the interest rate does not forecast future economic growth, which is a puzzle for the long-run risks model. Bansal and Yaron (2004) and Bansal, Kiku and Yaron (2012) argue that regressing the risk-free rate on consumption growth without including uncertainty will lead to a downward bias on the coefficient on growth. Beeler and Campbell (2012) argue that such a bias is small and cannot account for the lack of empirical relation. Importantly, this paper demonstrates that failing to include uncertainty is not responsible for this lack of predictability.12

While for parsimony the original long-run risks model only includes one stochastic volatility variable, in the model it is long-run, rather than short-run, volatility that matters for financial markets. Next, I empirically examine whether the interest rate appears to move more with short or long-term uncertainty and, consistent with the long-run risks model, find that the slower moving long-term uncertainty appears to be the dominant channel.

Similar to Bansal et al. (2014), I examine short-term economic uncertainty using the variance of month-to-month industrial production growth over the calendar year t+1. For the long-run measure, I examine the variance of month-to-month industrial production growth over the years t+1 to t+10. I utilize industrial production as it is available monthly with a time series that includes the entire baseline analysis period.

To examine whether the interest rate has a stronger relation with long-run or short-run volatility, I undertake an exercise similar to the one used before, but allow for long-run and short-run uncertainty so that each have a separate impact. Specifically, I run regressions of the form:  

(14)
rt=β0+β1Etgt+1+β2EtShort-RunUncertainty+β3EtLong-RunUncertainty.

In the analysis I utilize three different methods to create ex ante uncertainty measures. First, I simply utilize the ex post realized values. Second, I utilize lagged values of uncertainty to predict future by estimating  

(15)
Long-RunUncertaintyt+1;t+10=ϕgLong-RunUncertaintyt-9;t+ϵt+1,
 
(16)
Short-RunUncertaintyt+1=ϕgShort-RunUncertaintyt+ϵt+1.

Third, I also include the macro variables used to predict uncertainty from Bansal et al. (2014). Specifically, I add the year t values of consumption growth, growth in real per capita personal income, the price-to-dividend ratio, and return on the S&P 500 to Equations (15) and (16).

Table 14, panel A, presents regressions as described by Equation (14) utilizing ex post outcomes as the right-hand-side variables. Column 1 shows regressions with only the short-term uncertainty measure and finds a negative coefficient that is marginally significant. Column 2 uses the long-run measure, and the coefficient increases and becomes much more significant with a t-statistic of -3.95. Further the R2 increases from 0.05 to 0.34. The third column includes both measures of uncertainty. After both are included, the long-run measure retains its magnitude and significance, but the short-term uncertainty term switches sign and becomes insignificant with a t-statistic of 1.34. Thus, the real risk-free rate seems to predict long-term, rather than short-term, shifts in volatility.

Table 14

Short-run versus long-run uncertainty

A. Ex post 
Et[gt+1−0.126 0.153 0.183 
 (−0.19) (0.33) (0.41) 
Short-run uncertainty −16.656  9.969 
 (−1.82)  (1.34) 
Long-run uncertainty  −61.529 −71.893 
  (−3.95) (−3.84) 
Constant 0.020 0.043 0.042 
 (1.61) (3.40) (3.37) 
 
Observations 76 76 76 
Adjusted R2 0.05 0.34 0.34 
 
B. Time series 
 
Et[gt+1−0.272 0.204 0.177 
 (−0.46) (0.35) (0.31) 
Short-run uncertainty −38.295  −3.834 
 (−2.43)  (−0.24) 
Long-run uncertainty  −54.560 −52.169 
  (−3.03) (−2.47) 
Constant 0.044 0.040 0.042 
 (2.93) (3.31) (2.94) 
 
Observations 76 76 76 
Adjusted R2 0.09 0.23 0.22 
 
C. VAR 
 
Et[gt+1−0.475 −0.059 0.169 
 (−0.83) (−0.11) (0.28) 
Short-run uncertainty −29.530  18.345 
 (−2.83)  (1.05) 
Long-run uncertainty  −52.260 −69.871 
  (−3.05) (−2.56) 
Constant 0.038 0.042 0.035 
 (2.64) (3.18) (2.46) 
 
Observations 76 76 76 
Adjusted R2 0.07 0.18 0.18 
A. Ex post 
Et[gt+1−0.126 0.153 0.183 
 (−0.19) (0.33) (0.41) 
Short-run uncertainty −16.656  9.969 
 (−1.82)  (1.34) 
Long-run uncertainty  −61.529 −71.893 
  (−3.95) (−3.84) 
Constant 0.020 0.043 0.042 
 (1.61) (3.40) (3.37) 
 
Observations 76 76 76 
Adjusted R2 0.05 0.34 0.34 
 
B. Time series 
 
Et[gt+1−0.272 0.204 0.177 
 (−0.46) (0.35) (0.31) 
Short-run uncertainty −38.295  −3.834 
 (−2.43)  (−0.24) 
Long-run uncertainty  −54.560 −52.169 
  (−3.03) (−2.47) 
Constant 0.044 0.040 0.042 
 (2.93) (3.31) (2.94) 
 
Observations 76 76 76 
Adjusted R2 0.09 0.23 0.22 
 
C. VAR 
 
Et[gt+1−0.475 −0.059 0.169 
 (−0.83) (−0.11) (0.28) 
Short-run uncertainty −29.530  18.345 
 (−2.83)  (1.05) 
Long-run uncertainty  −52.260 −69.871 
  (−3.05) (−2.56) 
Constant 0.038 0.042 0.035 
 (2.64) (3.18) (2.46) 
 
Observations 76 76 76 
Adjusted R2 0.07 0.18 0.18 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by industrial production. Short-term uncertainty is the variance of monthly industrial production innovations in year t+1, and long-term uncertainty is through year t+10. Forecasts of growth are made from an ARMA(1,1)-GARCH(1,1) time-series model. Panel A uses ex post observed uncertainty. Panel B forecasts uncertainty using lagged values of the variables. Panel C repeats the analysis of panel B, but adds lagged values of consumption growth, the return and price/dividend ration of the S&P 500, and the change in real per capita disposable income. Standard errors are Newey-West using three lags. The top number is the coefficient, and the lower number in parentheses is the t-statistic.

The results in panel A are ex post realizations and could not be forecast at the time. Panels B and C utilize time-series models to predict the uncertainty estimates. In panel B lagged values, as described by Equations (15) and (16), are used to predict the short and long-run uncertainty. In panel C macro variables are added to these models to predict short-run and long-run uncertainty. Both panels B and C yield materially similar results to panel A. It appears the robust relation between uncertainty and the interest rate is driven by slow-moving long-run volatility of growth rather than short-run fluctuations.

8. Conclusion

Theory predicts that there is a relation between the interest rate, economic uncertainty, and economic growth. For the most part, the predicted relation between economic uncertainty and interest rates has not been emphasized. For example, a recent survey by the Council of Economic Advisors (2015) stresses the broad importance of understanding the real interest rate and identifies nine major factors that determine its level.13 These nine factors do not include the economic uncertainty examined here.

This paper demonstrates that economic uncertainty has a strong relation with the risk-free interest rate. This relation holds using a variety of specifications, datasets, and models. The results imply that analyses of the level of interest rates that leave out uncertainty may be seriously incomplete.

I would like to thank two anonymous referees, Kenneth Ahern, Wayne Ferson, Stefano Giglio, Chris Jones, Scott Joslin, Stavros Panageas, Ralitsa Petkova, Kelly Shue, David Solomon, Andreas Stathopoulos, Michael Weber, Fernando Zapatero, and seminar participants at the European Finance Association 2013 meetings, Chicago Booth, and the University of Southern California for helpful comments and suggestions.

1 For a recent survey, see Neely and Rapach (2008).
2 The December rate is chosen because it is the last value known when the growth forecasts are made. For the three-month interest rate, Table A1 shows results are not materially different using the January rate, using the compounding the rate from March, June, September, and December, forecasting the annual rate compounded from January, April, July, and October using an ARMA(1,1) process, or simply using the ex post realized rate.
3 These are the same data sources as used by Shiller (1989), though Shiller uses the ex post realized rate of rolling over in January and July, while I use the December rate annualized to avoid any forward-looking bias.
4 In untabulated results I find materially similar results by calculating the real interest rate using the methodology of Beeler and Compbell (2012) and Bansal, Kiku, and Yaron (2012).
5 For a more complete overview of the VIX, see the CBOE white paper on its construction (available at www.cboe.com/micro/vix/vixwhite.pdf).
6 For literature about the leading index, see Stock and Watson (1989, 1991), Hamilton and Perez-Quiros (1996), and Crone (2003). For literature about the predictive power of consumer sentiment, see Carroll, Fuhrer and Wilcox (1994), Matsusaka and Sbordone (1995), and Howrey (2001).
7 The interest rate is based on the December average, where monthly data are available. When only annual data are available, the annual measure is used.
8Table A2 shows the results for the basic estimation of different ARMA lags and shows that lag choice does not materially change the results.
9 Similar results are obtained using the median value at time t and estimating the relation analogously to Equation (1).
10 In untabulated results I find the annual results are similar using either consumption or GDP.
11 In untabulated results, proxies for inflation risk factors suggested by long-run risks and habit models (e.g., the covariance of inflation with habit) do not materially impact the results.
12 While an elasticity of intertemporal substitution greater than one is key to the model, some papers have provided evidence against this assumption (e.g., Beeler and Campbell 2012; Hall 1988; Campbell 2003). For the long-run risks model to yield this negative relation, the elasticity of intertemporal substitution must be greater than one. Thus, within the context of the long-run risks model, the results in this paper are consistent with an elasticity of intertemporal substitution greater than one.
13 The report lists transitory factors of fiscal, monetary, and foreign exchange policies, inflation risk and term premium, and private-sector leverage. The report lists longer lived factors of global output and productivity growth, demographics, “savings glut,” supply of safe assets, secular stagnation, and tail risks. The last is closest to this paper, but tail risks and Knightian uncertainty are a different type of uncertainty not examined by this paper.
14 Section 2.3 examines in more detail issues surrounding the real versus nominal interest rates and inflation.
15 If the initial VAR model is nonstationary, the Yule-Walker estimator is used:  
Φ^YW=t=1Txt-x*¯xt-1-x*¯t=0Txt-x*¯xt-x*¯-1,
where x*¯=1n+1t=0Txt.

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Appendix A

Consider a simple discrete-time representative agent model with conditionally lognormal disturbances (as in Hansen and Singleton 1983, Ferson 1983, and Harvey 1988). The lognormal disturbances allow for a closed-form solution. Conditional lognormality allows for heteroscedasticity as a function of the conditioning information. The consumer receives an endowment and chooses to consume it or invest $Pi,j in one of i=1, … ,n securities with j=1, … k maturities. Ct is consumption at time t, and the subscript of the conditional expectation indicates the time of the information on which the agent conditions. The consumer’s problem is  

(A1)
maxCt,Pi,j,tt=0t=0δtE0UCt.
This yields the first-order conditions (FOC):  
(A2)
EtδjU'Ct+jU'Ct1+Ri,j,t-1=0.
For each i,j security, Ri,j,t is the real return on asset i over j periods from t to t+j.14 Now let utility take the form:  
(A3)
UC,α=C1-α-11-αlnCifα=1ifα>1.

Substituting the utility function into the FOC and then taking logs yields:  

(A4)
lnEtδjCt+jCt-α1+Ri,j,t=EtδjCt+jCt-α1+Ri,j,t+12VartδjCt+jCt-α1+Ri,j,t=0.

Table A1

Regressions of interest rates on measures of growth and uncertainty

A. Consumption 
 January Rolling over Forecast Ex post 
Et[gt+1−0.8987 −1.0770 −1.6764 −0.7590 
 (−0.87) (−1.19) (−1.81) (−0.75) 
Vart[gt+1−65.5601 −60.6195 −53.2050 −64.3222 
 (−3.64) (−2.94) (−2.14) (−3.73) 
Constant 0.0402 0.0426 0.0540 0.0368 
 (1.69) (1.97) (2.36) (1.60) 
 
Observations 77 76 77 77 
Adjusted R2 0.16 0.15 0.13 0.15 
R2 excluding Vart[gt+10.00 0.00 0.04 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.012 −0.010 −0.012 
 
B. GDP 
 
 January Rolling over Forecast Ex post 
 
Et[gt+10.2600 0.1779 −0.0516 0.2589 
 (1.27) (0.83) (−0.17) (1.28) 
Vart[gt+1−13.2788 −12.8230 −11.5736 −13.0921 
 (−11.68) (−9.89) (−5.63) (−11.98) 
Constant 0.0194 0.0208 0.0236 0.0190 
 (2.83) (3.20) (3.28) (2.84) 
 
Observations 77 76 77 77 
Adjusted R2 0.54 0.54 0.41 0.54 
R2 excluding Vart[gt+1−0.01 −0.01 0.02 −0.01 
Impact of 1 SD Vart[gt+1−0.023 −0.023 −0.020 −0.023 
 
C. Industrial production 
 
 January Rolling over Forecast Ex post 
 
Et[gt+10.0726 −0.0124 −0.2535 0.0619 
 (0.32) (−0.05) (−1.18) (0.28) 
Vart[gt+1−2.3568 −2.4382 −2.0355 −2.3114 
 (−3.02) (−2.99) (−2.66) (−3.01) 
Constant 0.0189 0.0207 0.0218 0.0186 
 (3.49) (3.69) (3.59) (3.56) 
 
Observations 77 76 77 77 
Adjusted R2 0.33 0.34 0.27 0.32 
R2 excluding Vart[gt+10.00 0.00 0.04 0.00 
Impact of 1 SD Vart[gt+1−0.018 −0.018 −0.016 −0.018 
A. Consumption 
 January Rolling over Forecast Ex post 
Et[gt+1−0.8987 −1.0770 −1.6764 −0.7590 
 (−0.87) (−1.19) (−1.81) (−0.75) 
Vart[gt+1−65.5601 −60.6195 −53.2050 −64.3222 
 (−3.64) (−2.94) (−2.14) (−3.73) 
Constant 0.0402 0.0426 0.0540 0.0368 
 (1.69) (1.97) (2.36) (1.60) 
 
Observations 77 76 77 77 
Adjusted R2 0.16 0.15 0.13 0.15 
R2 excluding Vart[gt+10.00 0.00 0.04 −0.01 
Impact of 1 SD Vart[gt+1−0.013 −0.012 −0.010 −0.012 
 
B. GDP 
 
 January Rolling over Forecast Ex post 
 
Et[gt+10.2600 0.1779 −0.0516 0.2589 
 (1.27) (0.83) (−0.17) (1.28) 
Vart[gt+1−13.2788 −12.8230 −11.5736 −13.0921 
 (−11.68) (−9.89) (−5.63) (−11.98) 
Constant 0.0194 0.0208 0.0236 0.0190 
 (2.83) (3.20) (3.28) (2.84) 
 
Observations 77 76 77 77 
Adjusted R2 0.54 0.54 0.41 0.54 
R2 excluding Vart[gt+1−0.01 −0.01 0.02 −0.01 
Impact of 1 SD Vart[gt+1−0.023 −0.023 −0.020 −0.023 
 
C. Industrial production 
 
 January Rolling over Forecast Ex post 
 
Et[gt+10.0726 −0.0124 −0.2535 0.0619 
 (0.32) (−0.05) (−1.18) (0.28) 
Vart[gt+1−2.3568 −2.4382 −2.0355 −2.3114 
 (−3.02) (−2.99) (−2.66) (−3.01) 
Constant 0.0189 0.0207 0.0218 0.0186 
 (3.49) (3.69) (3.59) (3.56) 
 
Observations 77 76 77 77 
Adjusted R2 0.33 0.34 0.27 0.32 
R2 excluding Vart[gt+10.00 0.00 0.04 0.00 
Impact of 1 SD Vart[gt+1−0.018 −0.018 −0.016 −0.018 

This table presents regressions of various interest rates in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1934 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively, using data from 1934 until 2010. The interest rates used are indicated by the column header. “January” uses the January interest rate annualized from year t+1. “Rolling over” uses the interest rate compounded from months 3, 6, 9, and 12 in year t. “Forecast” models the interest rate from months 1, 4, 7, and 10 as an ARMA(1,1) and uses the prediction for year t. “Ex post” uses the realized interest rate from months 1, 4, 7, and 10 in year t+1. Forecasts are made from an ARMA(1,1)-GARCH(1,1) time-series model. The row labeled “Impact of 1 SD Vart[gt+1]” is the standard deviation of the variance of growth, multiplied by the coefficient on the variance of growth. Standard errors are Newey-West using three lags. The top number is the coefficient, and the lower number in parentheses is the t-statistic.

Table A2

Alternative lag length regressions of the real interest rate on forecasts of growth and uncertainty

 Consumption GDP Industrial production 
AR(1): Vart[gt+1−80.421 −13.048 −3.182 
 (−4.19) (−14.69) (−5.35) 
MA(1): Vart[gt+1−102.257 −11.953 −3.113 
 (−4.33) (−10.17) (−5.15) 
ARMA(1,1): Vart[gt+1−88.060 −13.105 −3.037 
 (−4.24) (−13.36) (−5.31) 
ARMA(2,2): Vart[gt+1−43.088 −4.140 −2.461 
 (−4.44) (−2.98) (−4.31) 
ARMA(3,3): Vart[gt+1−60.614 −6.836 −2.935 
 (−2.89) (−4.78) (−3.94) 
ARMA(4,4): Vart[gt+1−55.377 −6.541 −2.949 
 (−2.97) (−4.23) (−4.92) 
ARMA(5,5): Vart[gt+1−52.121 −7.391 −3.014 
 (−2.97) (−4.31) (−4.77) 
 
Observations 71 71 71 
 Consumption GDP Industrial production 
AR(1): Vart[gt+1−80.421 −13.048 −3.182 
 (−4.19) (−14.69) (−5.35) 
MA(1): Vart[gt+1−102.257 −11.953 −3.113 
 (−4.33) (−10.17) (−5.15) 
ARMA(1,1): Vart[gt+1−88.060 −13.105 −3.037 
 (−4.24) (−13.36) (−5.31) 
ARMA(2,2): Vart[gt+1−43.088 −4.140 −2.461 
 (−4.44) (−2.98) (−4.31) 
ARMA(3,3): Vart[gt+1−60.614 −6.836 −2.935 
 (−2.89) (−4.78) (−3.94) 
ARMA(4,4): Vart[gt+1−55.377 −6.541 −2.949 
 (−2.97) (−4.23) (−4.92) 
ARMA(5,5): Vart[gt+1−52.121 −7.391 −3.014 
 (−2.97) (−4.31) (−4.77) 
 
Observations 71 71 71 

This table presents regressions of the three-month Treasury-bill rate in year t on forecasts of the log of one, plus expected economic growth and the variance of the growth in year t+1 using data from 1939 until 2010. Growth is measured by NIPA consumption, GDP, and industrial production, respectively. Forecasts are made using the time-series model indicated in the left column. Standard errors are Newey-West using three lags. The top value is the coefficient, and the lower value in parentheses is the t-statistic.

Rearranging this equation, solving for the interest rate, and assuming it is risk free (i.e., known at time t) yields  

(A5)
ln1+Ri,j,t=-jlnδ+αEtlnCt+jCt-α22VartlnCt+jCt.

Using external habit formation from Campbell and Cochrane (1999), the specification of the risk-free rate becomes  

(A6)
ln1+Ri,j,t=-jlnδ+αEtlnCt+jCt-α22VartlnCt+jCt1+λst2-α1-φst-S¯,
where st is the log of the surplus-to-consumption ratio and λ(st) is the sensitivity function. Campbell and Cochrane Campbell (1999) specify λ(st) so that the last two terms offset, yielding the interest rate:  
(A7)
ln1+Ri,j,t=-jlnδ+αEtlnCt+jCt-12αS¯2VartlnCt+jCt.

Thus, again, this is simply a linear combination of expected growth and the variance of expected growth and can be expressed in the form of Equation (1). In the original model expected growth and the variance are constant, so the risk-free rate is constant.

Another class of models that has a received much attention in recent years is the long-run risks models introduced by Bansal and Yaron (2004). Constantinides and Gosh (2011) show that the risk-free rate can be written as  

(A8)
ln1+Ri,j,t=A0,f+A1,fxt+A2,fσt2,
where A0,f, A1,f, and A2,f, are time-invariant parameters (see Constantinides and Gosh’s (2011) Appendix A2.2 for explicit definitions), xt is the latent state variable, and σt is the variance of the state variable’s innovation. Using the fact that  
(A9)
EtlnCt+jCt=xt+μc,
where μc is a constant, and defining the constant Af=A0,f+A1,f μc, one can write  
(A10)
ln1+Ri,j,t=Af+A1,fEtlnCt+jCt-A2,fVartlnCt+jCt.

Thus, the long-run risks model is also consistent with Equation (1).

Appendix B

The implementation of the mARM procedure (Amihud, Hurvich, and Wang 2009) in this paper follows closely to that described by Amihud and Hurvich (2004), the appendix of Avramov, Barras, and Kosowski (2013), and the appendix of Ferson, Nallareddy, and Xie (2013).

{xt} denotes a p-dimensional vector of predictors from t=0, , T, where for this paper p = 2 and includes forecasts of growth and uncertainty. Also, to make the notation consistent with Amihud, Hurvich, and Wang (2009), xt denotes the forecast made using data at time t for period t+1. Further, rt denotes the interest rate from t - 1 to t. The model is thus given by  

(A11)
rt=α+β'xt-1+ut,
 
(A12)
xt=Θ+Φxt-1+νt,
 
(A13)
xt=x1tx2t,Θ=Θ1Θ2,β=β1β2,νt=ν1tν2t,Φ=Φ11Φ12Φ21Φ22.

Amihud and Hurvich (2004) show that one can write  

(A14)
ut=φνt+et.

So the model with bias correction is estimated as  

(A15)
rt=α+βxt-1+φνt+ut.

To implement the estimate, the following procedure is used. Estimate the expression for xt utilizing a VAR(1) regression.15 This expression yields the preliminary estimate, Φ^ and Θ^, along with the covariance matrix from the residuals νt denoted Σ^v. To estimate the small sample bias, the Nicholls and Pope (1988) estimate is used:  

(A16)
bias^Φ^=EΦ^-Φ=1TΣ^vI-Φ^-1+ΦI-Φ^2-1+j=12λjI-λjΦ^-1Σ^x,
where I is a 2x2 identity matrix, λj is the jth eigenvalue of Φ^, and Σ^xis estimated using the formula vecΣ^x=I4-ΦΦ-1vecΣ^v.

Using these initial estimates (denoted Φ^0), an iterative procedure is used to construct estimates as follows. The new estimate of coefficients is Φ^1=Φ^0-bias^Φ^0; the new estimate of the constant is Θ^1=I-Φ^1Z¯ (where Z¯ is the sample mean); and a new estimate of Σ^v is obtained from the residuals. If the model is nonstationary, the iterations stop and values from that iteration are used. If not, the procedure repeats itself using the previous steps and estimates to construct the next iteration for a maximum of ten iterations. Taking the final values from this procedure, Equation (A15) is estimated.

To construct test statistics, the variance of β^ is calculated, where all terms with hats are the estimates from the final round of the iteration:  

(A17)
var^β^j=i=12φ^i2var^Φ^ij+i=12ki22φ^iφ^jcovΦ^ij,Φ^kj+SE^β^j,
where SE^β^j is the ordinary least-squares standard error from the augmented regression of Equation (A15). To implement the t-test, the test statistic is calculated as  
(A18)
t=β^jvar^β^j.