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BY 4.0 license Open Access Published online by De Gruyter June 7, 2022

A threshold model for the spread

Dimitris Hatzinikolaou and Georgios Sarigiannidis


Using annual data from two panels, one of 11 Eurozone countries and another of 31 OECD countries, we estimate a two-regime log-linear as well as a nonlinear model for the spread as a function of macroeconomic and quality-of-institutions variables. The two regimes, a high-spread and a low-spread regime, are distinguished by using a threshold, in accordance with the perceived “fair” value of the spread as a reference point. Our results suggest that government-bond spreads are regime-dependent, as most of the regression coefficients of the determinants of the spread are larger (in absolute value) in the high-spread regime than in the low-spread regime. That is, an improvement in the macroeconomic environment (e.g., lower unemployment, lower inflation, lower growth of the debt-to-GDP ratio, less macroeconomic uncertainty, higher growth of real GDP), and/or an improvement in the quality of institutions (e.g., less corruption) reduce the spread facing a country (by enhancing its creditworthiness) to a greater extent in high-spread situations than in low-spread situations. A possible explanation is that the demand for and the supply of loans are inelastic at higher than “fair” interest rates and elastic at lower rates.

JEL Classification: E44; F34; G01; G12; G15; H63

1 Introduction

In recent years, the deterioration of macroeconomic indicators (fiscal imbalances, government indebtedness, unemployment, etc.), in both developed and less developed countries, and the governments’ borrowing strategies have given rise to concerns about the risks posed for individual countries as well as for the global financial system. Some studies question the notion that spreads can be explained only by macroeconomic indicators in one state of the world, however, suggesting that other factors might be affecting spreads, such as the perception of sovereign risk at various spread levels (e.g., Gomez-Bengoechea and Arahuetes 2019; Haugh, Ollivaud, and Turner 2009). Furthermore, when assessing risk, it appears that some variables, which determine a country’s creditworthiness, e.g., macroeconomic and quality of institutions indicators, are more important in periods of high risk exposure (hence high spreads), and less important in periods of low risk exposure (low spreads).

This paper is an empirical analysis that aims to provide new insights about the link between macroeconomic indicators and sovereign bond yield spreads, assuming two states of the world. We consider a decision maker in charge of financing his/her government’s spending program by borrowing from the markets, which charge him/her an interest rate in accordance with the prevailing macroeconomic conditions and the quality of institutions, serving as indicators of the borrower’s credibility. We assume that he/she believes that the country’s financial condition is sufficient for a loan of a specified amount at a “fair” interest rate, known to both parties. We then ask: How would the borrower respond if the markets offered him/her an interest rate (r) that is (i) higher than the subjective “fair” rate, a state of affairs which we call high-spread regime, or (ii) lower than or equal to the “fair” rate, a state we call low-spread regime?

Here, we are contemplating the possibility that the borrower’s demand for loans is inelastic at higher than “fair” interest rates and elastic at lower rates. Such behavior can be described by the log-inverse model, L = exp(β0 + β1/r), where L is the quantity of loan demanded and β1 > 0. The interest elasticity of demand in this case is −β1/r < 0. As a consequence, a given increase in the demand for loans raises the spread by more at the high-spread regime than at the low-spread regime. This is our “threshold-value hypothesis,” the novelty of the paper, in which the “reference point” is assumed to be the “threshold” that separates the two regimes.

We make this hypothesis operational by defining a dummy variable, I, and its complement, 1 – I; multiplying each with the determinants of the spread, thus creating interaction variables that will allow the effect of each determinant to differ from regime to regime; and testing the hypothesis of equality of the coefficients of the corresponding interaction variables in our regressions for the spread. If the coefficients of these interaction variables in the high-spread regime turn out to be larger (in absolute value) than their counterparts in the low-spread regime, this will lend support to our hypothesis, as providing evidence that a given change in a determinant of the spread that shifts the demand for loans, thus leading to a new equilibrium in the loan market, influences the spread by more in the high-spread regime than it does in the low-spread regime.

To illustrate this nonlinearity, consider Figure 1, which is consistent with the log-inverse model for the demand for loans given above. Assume that initially our decision maker borrows an amount L0 at an interest rate r0, but then an increase in his/her demand for loans occurs, say, because unemployment (a shift variable hidden in the intercept, β0) has increased, and he/she needs funds to implement a “lean against the wind” policy. Thus, at the initial interest rate (r0), which prevailed when quantity demanded was L0, his/her quantity demanded increases from L0 to L1 and there occurs an excess demand for loans, which pushes the interest rate up. After all, during periods of high unemployment, the country’s economy is weaker and the markets require an additional premium to compensate for their risk exposure. Now, a large increase in the interest rate (the high-spread regime), e.g., Δri = r1ir0, is associated with an inelastic demand of the decision maker, whereas a small increase (the low-spread regime), e.g., Δre = r1er0, where Δri > Δre, is associated with an elastic demand. (The superscripts i and e stand for “inelastic” and “elastic” demand, respectively.)

Figure 1: 
Elasticity of demand for loans and the effect of borrowing on the spread.

Figure 1:

Elasticity of demand for loans and the effect of borrowing on the spread.

The above argument can be reinforced by assuming that the supply of loans is also inelastic at higher than “fair” interest rates and elastic at lower rates. Figure 2 shows the case of an initial and a new equilibrium (after a shift in demand) that occur at low rates, where both demand and supply are highly elastic.[1] The figure shows an initial equilibrium at point A, where the interest rate is r0, and then an increase in the demand for loans, say, because of an increase in the rate of unemployment, so equilibrium moves to point B. Because the supply and demand curves are assumed to be highly elastic at low interest rates, a small increase in the interest rate, say, from r0 to r1, suffices to restore equilibrium after a large increase in demand.

Figure 2: 
Competitive equilibrium with highly elastic supply and demand curves for loans at low interest rates, where only a negligible increase in the interest rate is required to restore equilibrium after a large increase in demand.

Figure 2:

Competitive equilibrium with highly elastic supply and demand curves for loans at low interest rates, where only a negligible increase in the interest rate is required to restore equilibrium after a large increase in demand.

Using annual data from two panels, one of 11 Eurozone countries and another of 31 OECD countries, we estimate a two-regime threshold model in accordance with Hansen (1999) [2] and Cassou, Shadmani, and Vázquez (2017). This simple model serves our purpose satisfactorily, which is to test our “threshold-value hypothesis” via the interaction variables defined above, so we did not pursue more sophisticated approaches, such as a Markov switching model, a “panel smooth transition regression model” (González et al. 2017; Hmiden and Cheikh 2016), or a model that relates the threshold more directly to fundamentals, such as public debt (Hadzi-Vaskov and Ricci 2019).

We employ a log-linear as well as a nonlinear spread equation. The basic model is derived in Feder and Just (1977a, 1977b), Edwards (1984), and elsewhere, and considers the spread as a function of the probability of default, which is itself a function of a set of macroeconomic and other variables. In accordance with the literature, this set includes the growth rate of GDP, the unemployment rate, the inflation rate, the primary budget surplus, the growth rate of debt to GDP, an uncertainty index, the real exchange rate, a risk-free interest rate, a United States (US) long-term interest rate, private investment, public investment, government effectiveness, control of corruption, and political stability (Afonso 2003; Afonso and Jalles 2019; Cantor and Packer 1996; Hadzi-Vaskov and Ricci 2019; Hmiden and Cheikh 2016).

We use two empirical definitions of the “fair” value of the spread, denoted as M and R, which are described in detail in Section 2. Briefly, according to definition M, there is only one “fair” value of the spread for all the countries in the sample in year t, and that is the average spread in year t; whereas, according to definition R, there are as many “fair” values as rating classes, and the “fair” value for each country is the average spread of the associated rating class that the country belongs to in year t. In both definitions, we use the simple (instead of a weighted) average, since there is no substantial variation across countries. We also use two estimation methods, least squares and the generalized method of moments (GMM). With two functional forms, two panels of data, two definitions of the “fair” spread, and two estimation methods, we end up estimating 16 regressions. To confirm or reject our assertions and to judge the reliability of the 16 regressions, we use a battery of tests.

Our results broadly support our “threshold-value hypothesis,” as most of the coefficients of the interaction variables defined earlier are larger (in absolute value) in the high-spread regime than their counterparts in the low-spread regime. More specifically, we find that macroeconomic variables (e.g., unemployment rate, inflation rate, growth rate of real GDP, growth rate of the debt-to-GDP ratio, and an uncertainty index), as well as some quality of institutions indices (e.g., a corruption index), all of which are important determinants of a country’s creditworthiness, have a greater effect on the spread in high-spread situations than in low-spread situations. Incorporating this hypothesis into our model and testing it aims at providing the policy maker with a more adequate explanation of the mechanism that determines the spread.

Section 2 specifies the model, Section 3 describes the data and discusses their stochastic properties, Section 4 estimates and tests the 16 regressions and discusses the results, and Section 5 concludes.

2 The model

According to Feder and Just (1977b), Eaton and Gersovitz (1981), and Sachs (1983), among others, the spread over LIBOR charged on Eurodollar loans taken by a country reflects its probability of default, p. Thus, following Feder and Just (1977b), Edwards (1984), Bassat and Gottlieb (1992), and Ozdemir (2004), we assume that

(1) s = p / ( 1 p ) γ ,

where γ = 1 + r*, and r* is a risk-free interest rate. Edwards (1984) derives Eq. (1) by assuming perfectly competitive financial markets and risk-neutral lenders. The lender is indifferent between gaining the return 1 + r with probability 1 – p and the risk-free return 1 + r* with certainty. Equilibrium requires that the following condition hold:

(2) ( 1 p ) ( 1 + r ) = 1 + r * .

Solving this equation for r, subtracting r* from both sides of the resulting equation, and setting s = rr* and γ = 1 + r* yields Eq. (1).

The (subjective) probability p is not observable, but can be estimated by assuming that it is a function of variables collected in a vector x, say, p(x) (Feder and Just 1977b). Since p(x) must be bounded between zero and one for all choices of x, the most widely used functional form for p(x) is the logistic form (Cox 1970), which gives

(3) p ( x ) / [ 1 p ( x ) ] = exp β 0 + Σ j = 1 k β j x j .

Combining (1) and (3) yields the following nonlinear equation for the spread:

(4) s = γ exp β 0 + Σ j = 1 k β j x j .

Taking logarithms in (4); adjusting the notation for panel data; and assuming that γ is a time-varying variable whose value in year t applies to all the countries in the sample, whereas β0 is constant across time and countries; yields the log-linear equation

(5) log s i t = β 0 + Σ j = 1 k β j x i t , j + log γ t

Thus, denoting the threshold value as s t *, the two-regime log-linear model is

(6) log s i t = β 0 + Σ j = 1 k β j x i t , j + α log γ t , s i t > s t * , log s i t = β 0 + Σ j = 1 k β j x i t , j + α log γ t , s i t s t * ,

where all coefficients are allowed to vary from regime to regime.

Again, we use two definitions of the “fair” spread. First, for a given year t, we take the spreads facing all the countries in the sample and calculate their mean, denoted as s t * , which is the same for all the countries in the sample. This is the definition used in Eq. (6); in what follows, it is denoted by M.

Second, we sort the countries into rating classes, depending on their credit rating (provided by Moody’s), and measure the “fair” spread facing country i in year t by the average spread of the associated rating class that country i belongs to in year t. This “fair” value, denoted as s it * , is indexed by i, because in a given year t, the various countries in the sample belong to different rating classes. For example, if country i is rated AAA in year t, then s it * is the average spread of all the countries rated AAA in year t, but in the same year countries in other rating classes will be faced with a different “fair” spread.[3] In what follows, this rating-based definition is indicated by R.

An advantage of using definition R is that country ratings are taken into account. There are two problems with this definition, however. First, in terms of Figure 1, it implies that the demand functions for funds of two countries that belong to different rating classes exhibit the same curvature. Thus, for example, if these two countries end up paying the same spread (say, because the better rated country experiences negative shocks for a period of time), their demand functions for funds may falsely exhibit the same nonlinearity. Second, it introduces an endogeneity problem noted by Hansen (1999, pp. 346 and 356), as the threshold becomes an endogenous variable. For if there is a negative shock in country i that raises its spread, the average spread in its rating class will rise, especially if the number of countries in that class is small, and this may cause some of the other countries in that class to be placed in a wrong regime. Thus, as will be seen shortly, this problem renders all of the explanatory variables used in our regressions endogenous. In contrast, when we use definition M, this problem is expected to be much less severe, as an increase in country i’s spread will not affect noticeably the average spread of the entire panel in year t.[4]

A threshold regression emerges by combining the two regimes in Eq. (6), using the dummy variable I it , where I it = 1 if s it > s t * and I it = 0 if s it s t *, and the country dummies D1, …, DN-1, to capture fixed country-specific effects (ignoring time effects for the moment), and by adding an error term, ε it , which reflects both supply and demand shocks, since Eq. (1) is derived from the equilibrium condition (2). The result is

(7) log s i t = β 0 I i t + β 0 ( 1 I i t ) + Σ i = 1 N 1 θ i D i + Σ j = 1 k β j I i t x i t , j + β j ( 1 I i t ) x i t , j + α I i t log γ t + α ( 1 I i t ) log γ t + ε i t .

The logarithmic transformation could potentially destroy the consistency of the estimates, however,[5] so we also estimate the nonlinear model, Eq. (4), the threshold version of which can be written as

(8) s i t = γ t exp β 0 I i t + β 0 ( 1 I i t ) + Σ i = 1 N 1 θ i D i + Σ j = 1 k β j I i t x i t , j + β j ( 1 I i t ) x i t , j .

When Eq. (8) was estimated, there was evidence of serial correlation, which could be interpreted as a sign of model misspecification, so we modify Eq. (8) to incorporate the partial adjustment mechanism. Let s it d denote the desired level of the spread, which is not directly observable, and assume that if there is a shock to it, a complete adjustment of the actual spread (s it ) to its desired level may not be achieved in a single period, because of technological constraints, institutional rigidities, lags in the publication of data, etc., but only a fraction δ of the ratio s i t d / s i t 1 may be translated into a change in the actual spread ratio s it /sit−1, where δ is the “speed of adjustment” parameter. The closer the value of δ is to 1 the faster the speed of adjustment. That is, s i t / s i t 1 = s i t d / s i t 1 δ , 0 ≤ δ ≤ 1 (Kmenta 1971, pp. 476–477). Modifying this equation, to allow δ to vary from regime to regime, yields

(9) s i t / s i t 1 = s i t d / s i t 1 δ 1 I i t + δ 2 1 I i t .

Replacing sit with sitd on the left-hand side of (8), substituting the result into (9) for sitd, and adding an error term, we obtain

(10) s i t = exp δ 1 I i t + δ 2 ( 1 I i t ) β 0 I i t + β 0 ( 1 I i t ) + Σ i = 1 N 1 θ i D i + Σ j = 1 k β j I i t x i t , j + β j ( 1 I i t ) x i t , j γ t δ 1 I i t + δ 2 1 I i t s i t 1 1 δ 1 I i t δ 2 1 I i t + ε i t .

Note that in six out of the eight cases where we estimate Eq. (10) the serial correlation problem has disappeared (see Regressions 5, 6, and 13–16 in Tables 4 and 6). Note also that for δ1 = δ2 = 1 (immediate adjustment), Eq. (10) reduces to (8).

3 The data

We use annual data from two unbalanced panels. The first, Panel 1, considers 11 Eurozone countries over the time period 1997–2014, namely Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Portugal, and Spain. This group of countries has been used in various papers, such as Barrios et al. (2009). Its selection is based on data reliability and availability limitations, mainly for the Baltic countries, which joined the Eurozone only recently. The macroeconomic data for this group of countries were collected from the Eurostat Database.

As a robustness check, we also use another panel of 31 OECD countries, Panel 2, over the time period 1996–2014. In addition to the above 11 Eurozone countries, Panel 2 includes Australia, Canada, Chile, Czech Republic, Denmark, Hungary, Iceland, Israel, Japan, Korea, Mexico, New Zealand, Norway, Poland, Slovak Republic, Slovenia, Sweden, Switzerland, United Kingdom, and the United States.

The variables used in our analysis are as follows: (1) the spread facing country i, which is measured as the difference between its 10-year government-bond yield and the Libor; (2) GAMMA = 1 + r* and LGAMMA = log(1 + r*), where r* is assumed to capture shocks that occur in international markets; (3) the inflation rate (INFL); (4) the growth rate of the debt-to-GDP ratio (DEBTGRO);[6] (5) the growth rate of real GDP (GDPGRO); (6) the primary budget surplus (PRIMBS); (7) the unemployment rate (UR); (8) public investment (PUBINV); (9) a volatility index (VIX), which serves as an indicator of uncertainty; (10) the real exchange rate (REER_WB), an increase in which means real depreciation of the home currency; (11) the 10-year US government bond yield (LONG_US); and (12) several indices of quality of institutions, like control of corruption (CORRU), political stability (POLSTAB), government effectiveness (GOVEFFECT), etc. Higher values of these indices imply higher quality of institutions. To facilitate the interpretation of the regression coefficients reported in Tables 36, we report the sample means of these variables in Table 1.

Table 1:

Sample means of the variables.

Panel 1: 11 Eurozone countries Panel 2: 31 OECD countries
(1997–2014) (1996–2014)
Variable Mean Variable Mean Variable Mean Variable Mean
LIBOR 0.0177 LONG_US 4.1400 LIBOR 0.01920 LONG_US 4.2611
LSLIBOR −4.2648 REER_WB 99.7458 LSLIBOR −4.0558 REER_WB 99.4160
GAMMA 1.0346 CORRU 1.5976 GAMMA 1.0359 CORRU 1.3583
LGAMMA 0.0340 POLSTAB 1.0385 LGAMMA 0.0351 POLSTAB 0.7878
INFL 0.0201 RLAW 1.5453 INFL 0.0288 RLAW
UR 0.0819 PRINV 132.7896 UR 0.0750 PRINV
GDPGRO 0.0166 PUBINV 22.4448 GDPGRO 0.0242 PUBINV 30.5815
DEBTGRO 0.01705 GOVEFFECT 1.6005 DEBTGRO 0.0212 GOVEFFECT 1.3943
PRIMBS −0.0301 DUMMYM 0.5278 PRIMBS −0.0212 DUMMYM 0.6197
VIX 21.4689 DUMMYR 0.5966 VIX 21.2042 DUMMYR 0.4720

We also construct interaction terms after defining the dummy variable I1 as I1 it = 1 – I it , e.g., IUR = I × UR, I1UR = I1 × UR, ICORRU = I × CORRU, I1CORRU = I1 × CORRU, etc. From these definitions, it is obvious that, to the extent that the dummy variables I it and I1 it are endogenous (see our discussion on the definitions M and R for the “fair” spread in Section 2) so are these interaction variables; and that is true even for variables that can be taken as exogenous in this paper, such as CORRU. This problem exacerbates the already existing endogeneity problem, since the macroeconomic determinants of the spread are inherently endogenous. Consider, for example, the rate of growth of the debt-to-GDP ratio (DEBTGRO). A positive shock in Eq. (7), ε it > 0, which causes the spread to rise, it also causes DEBTGRO to rise, so DEBTGRO is expected to be correlated with ε it . Thus, GMM estimates (see Tables 5 and 6) should be considered more reliable than the least-squares ones (Tables 3 and 4). This discussion implies that if we use the above variables (IUR, I1UR, ICORRU, etc.) as instruments, we should lag them at least by one time period.

The sources of the data are as follows: The 10-year government bond yields, the Libor rate over Euro, and the volatility index were obtained from the Federal Reserve Bank of St. Louis. The risk-free rate (r*), proxied by Germany’s long-run interest rate, was obtained from the European Central Bank (ECB Monthly Bulletin 2014). The quality of institutions indicators and the real exchange rate were obtained from the World Bank, whereas the remaining series from the Eurostat.

Next, we perform the following unit-root tests: (1) Levin, Lin, and Chu (2002) (LLC); (2) Breitung (2000); (3) Im, Pesaran, and Shin (2003) (IPS); (4) Fisher-type Augmented Dickey-Fuller (ADF) test; and (5) Fisher-type Phillips-Perron (PP) test (Maddala and Wu 1999, and Baltagi 2001, pp. 240–241). The results are reported in Table 2. Note, first, that the LLC and Breitung tests require “strongly-balanced” panels, so these two tests do not provide any results for the variables that do not satisfy this requirement. Second, for each variable, stationarity is supported by at least two tests, so we take all the variables used in our regressions to be I(0). Third, depending on the deterministic components included in the testing regression, each test produces several statistics; we take the one that most strongly supports stationarity. Fourth, as we noted earlier, the tests suggest that the debt-to-GDP ratio and its logarithm are I(1), so we do not use these variables. Finally, the tests suggest that the first differences are I(0).

Table 2:

Unit-root tests.

Test LLC Breitung IPS Fisher-ADF Fisher-PP I(0) or I(1)?
LIBOR −6.6*** 173.2*** (c) 155.8*** (c) I(0)
LSLIBOR 92.5*** (c) 177.2*** (c) I(0)
GAMMA 0.4 −6.1*** −8.7*** 37.2 145.5*** I(0)
LGAMMA 0.7 −6.0*** −8.5*** 33.1 137.9*** I(0)
INFL −5.8*** (c) −3.8*** −8.0*** 192.2*** (c) 296.3*** (c) I(0)
UR −8.4*** −2.0** (c) −1.5* 153.3*** 76.5 (c) I(0)
GDPGRO −8.5*** (c) −9.1*** (c) −9.3*** 186.3*** (c) 231.4*** (c) I(0)
DEBTGDP 0.6 74.5* (c) 62.3 (c) I(1)
LDEBTGDP 0.1 70.3 (c) 70.4 (c) I(1)
DEBTGRO −10.9*** 140.2*** 715.3*** (c) I(0)
PRIMBS −5.3*** 141.5*** (c) 114.4*** (c) I(0)
VIX −11.4*** (c) −7.9*** −3.8*** 209.4*** (c) 99.1*** (c) I(0)
LONG_US −10.9*** −6.7*** −7.4*** 158.9*** 108.1*** I(0)
REER_WB −3.4*** (c) −1.2 (c) −4.3*** 103.0*** (c) 103.7*** (c) I(0)
CORRU −3.5*** 56.8 162.9*** I(0)
POLSTAB −5.5*** 102.9*** (c) 175.3*** I(0)
RLAW −7.3*** 88.2*** 79.7*** (c) I(0)
PRINV −5.0*** (c) −0.5 −2.8 *** 111.2*** (c) 77.1*** (c) I(0)
PUBINV −5.1*** (c) −2.7*** (c) −4.3*** 109.7*** (c) 94.0*** (c) I(0)
GOVEFFECT −9.4*** −2.5*** (c) −5.0*** 189.0*** 158.0*** I(0)
DUMMYM −7.7*** −8.3*** (c) −7.9*** 150.9*** (c) 196.0*** (c) I(0)
DUMMYR −7.6*** 234.5*** (c) 267.1*** (c) I(0)

  1. (1) ***, **, and * indicate statistical significance at the 1-, 5-, and 10-percent level, respectively, assuming one-sided alternatives; (2) the indication (c) next to the value of a test statistic means that only a constant was included in the deterministic components of the regression, but not a trend; whereas no indication at all means that both a constant and trend were included; (3) the reported statistic for the Fisher-type Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests is inverse chi-squared (denoted by P), whereas that for the IPS test is the “Z-t-tilde-bar” test; (4) blank spaces occur because the LLC and Breitung tests require strongly balanced panels, and some of our variables do not satisfy this requirement; (5) in each cell of the table, we report the figure that most strongly supports the stationarity hypothesis; (6) for a given variable, if at least two tests support stationarity, the variable is taken to be I(0); otherwise it is deemed I(1); (7) the figures reported in this table have been derived by the econometric program STATA.

4 Empirical results

Tables 36 report our empirical results for the two Equations, (7) and (10), two panels of data, Eurozone and OECD, two estimation methods, least squares (with robust standard errors to heteroscedasticity and to serial correlation) and GMM, and two definitions for the “fair” spread, M and R, 16 regressions altogether. The log-linear regressions (1–4 and 9–12) include both country and time dummies (to allow for the possibility that the years from 2008 onward might have had a special impact on the spread, because of the financial crisis), whereas the nonlinear ones (5–8 and 13–16) include only country dummies, to avoid convergence problems.

Table 3:

OLS estimation of the log-linear model, Eq. (7), fixed country and time effects, Dep. variable: log of spread over Libor, M = Mean spread as threshold, R = Rating-based threshold.

Explanatory variables & tests Panel 1: 11 Eurozone countries Panel 2: 31 OECD countries
Regr. 1 (M) Regr. 2 (R) Regr. 3 (M) Regr. 4 (R)
Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat)
I −4.51*** (−25.9) −5.19*** (−21.0) −0.03 (−0.2) −0.60 (−1.1)
ILGAMMA −32.79*** (−9.1) −12.67*** (−2.9) −11.82*** (−5.2) −13.26*** (−5.0)
IINFL 12.18*** (6.4) 15.00*** (5.9) 1.28 (0.4) 2.49 (1.1)
IDEBTGRO 1.05*** (4.0) 1.46*** (5.1) −0.29 (−0.9) −0.09 (−0.4)
IUR 6.16*** (9.4) 8.05*** (8.8) 5.17*** (4.8) 5.02*** (4.3)
IVIX 0.04*** (5.1) 0.02*** (2.6) −0.001 (−0.2) −0.00 (−0.2)
IGDPGRO −4.93*** (−4.6) −8.65*** (−6.3) −8.70*** (−4.3) −7.08*** (−4.0)
IPRIMBS 0.72 (1.5) −0.37 (−0.6) −1.98*** (−2.4) −2.24** (−1.8)
IREER_WB 0.09 (1.9) −0.00 (−0.2) −0.005** (−1.7) −0.004** (−1.6)
IPUBINV −0.001 (−0.7) −0.004*** (−3.5) −0.009*** (−7.1) −0.010*** (−9.1)
IPRINV 0.00 (0.0) 0.00 (0.8)
IPOLSTAB 0.04 (0.4) 0.22 (1.1) 0.13 (0.9) 0.04 (0.5)
I1 −3.10*** (−3.7) −4.20*** (−20.5) −0.15 (−0.8) −0.58 (−0.7)
I1LGAMMA −3.57 (−0.8) 1.05 (0.1) −13.09*** (−3.9) −2.00 (−1.0)
I1INFL 1.34 (0.3) −2.55 (−0.6) −2.91 (−0.8) −3.03 (−1.0)
I1DEBTGRO −1.62 (−1.5) −0.50 (−0.4) 0.20 (0.8) 0.20 (0.5)
I1UR 1.53 (0.8) 1.74 (0.9) 3.93*** (2.7) 4.61*** (3.2)
I1VIX 0.01*** (2.3) 0.02** (1.9) 0.02*** (2.8) 0.00 (0.2)
I1GDPGRO −9.31*** (−5.0) −7.88*** (−4.4) −4.74*** (−3.5) −9.22*** (−4.0)
I1PRIMBS −5.55*** (−3.4) −5.95*** (−4.5) −2.84*** (−2.6) −1.83** (−1.9)
I1REER_WB −0.01** (−1.7) 0.00 (0.2) −0.01*** (−3.3) −0.01*** (−4.1)
I1PUBINV −0.002 (0.5) −0.001 (−0.4) −0.009*** (−10.5) −0.008*** (−10.4)
I1PRINV −0.00* (−1.6) −0.00 (−0.4)
I1POLSTAB −0.38*** (−4.1) −0.59*** (−5.5) −0.04 (−0.3) −0.06 (−0.7)
Usable obs. (n) 113 111 476 471
Centered R2 0.95 0.93 0.64 0.66
H0: One regime χ 11 2 = 605 *** (0) χ 10 2 = 295 *** (0) χ 11 2 = 31 *** (0) χ 6 2 = 744 *** (0)
RESET χ 1 2 0.0 (0.50) 0.02 (0.89) 0.4 (0.54) 0.5 (0.50)
DWH χ 12 2 = 36 *** (0.0) χ 12 2 = 34 *** (0.0) χ 13 2 = 73 *** (0.0) χ 11 2 = 48 *** (0.0)
H0: no serial corr. t54 = 0.8 (0.40) t55 = 3.1*** (0.00) t390 = 6.9*** (0.00) t385 = 5.4*** (0.00)
H0: no break in 2008 χ 8 2 = 26 . 5 * * * (0.00) χ 8 2 = 46 . 3 * * * (0.00) χ 3 2 = 1.4 (0.71) χ 3 2 = 1.7 (0.60)
χ 14 2 = 91 . 2 * * * (0.00) χ 13 2 = 75 . 6 * * * (0.00) χ 13 2 = 97 *** (0.00) χ 6 2 = 13.6 ** (0.04)

  1. (1) A line separates the two regimes; (2) the numbers in parentheses next to the coefficient estimates are t-statistics; those next to the test statistics are p-values; (3) the routine LINREG of WinRATS 9.1 was used with the option Eicker-White standard errors; (4) shaded coefficients refer to variables that were dropped, as insignificant; (5) A dash in a cell means that no estimate is available due to insufficient data on that variable; (6) Note (1) to Table 2 applies here, too.

Table 4:

Estimation of Eq. (10) by NLLS, fixed country-specific effects only, Dep. variable: Spread over Libor, M = Mean spread as threshold, R = Rating-based threshold.

Explanatory variables & tests Panel 1: 11 Eurozone countries Panel 2: 31 OECD countries
Regr. 5 (M) Regr. 6 (R) Regr. 7 (M) Regr. 8 (R)
Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat)
I −7.22*** (−6.6) −6.57*** (−16.0) −3.18*** (−10.2) −2.76*** (−13.0)
IINFL 28.94*** (3.6) 26.26*** (2.7) 13.14*** (2.9) 13.22*** (3.1)
IDEBTGRO 1.33*** (3.3) 1.06*** (4.6) 0.90** (2.1) 1.06*** (6.4)
IUR 12.92*** (11.5) 9.69*** (13.3) −5.57 (−1.5) −24.10 (−1.4)
IVIX 0.11** (2.2) 0.13*** (4.3) 0.04* (1.4) 0.30 (1.2)
IGDPGRO −11.86*** (−3.6) −8.11*** (−3.9) −16.13*** (−7.3) −14.62*** (−7.5)
IPRIMBS 49.42 (0.7) −1.18 (−0.7) 17.84 (1.4) −21.40 (−1.1)
IPUBINV 0.001 (0.3) −0.010 (−0.7) −0.002*** (−2.9) −0.002*** (−8.0)
IPOLSTAB −0.47** (−1.8) −7.48 (−0.9) −0.33*** (−5.0) −0.28*** (−6.0)
ILONG_US −0.01 (−0.1) −0.38*** (−5.3) −0.24*** (−2.9) −0.15*** (−3.1)
ICORRU 0.31 (0.7) 3.06 (0.9) −0.05 (−0.1) −0.71 (−0.6)
δ ̂ 1 0.51*** (3.2) 0.50*** (8.3) 0.37*** (4.9) 0.48*** (5.9)
I1 −5.53*** (−24.5) −6.81*** (−4.7) 2.31 (0.3) −104.15 (−0.6)
I1INFL 16.26*** (2.4) 103.01 (0.7) 252.12 (0.4) −9.62 (−0.1)
I1DEBTGRO −4.52 (−1.0) 10.23 (0.9) 27.84 (0.4) 8.93 (0.9)
I1UR 5.52** (2.2) 6.82 (1.2) 20.93 (0.4) 69.84 (0.8)
I1VIX 0.01 (0.5) 0.10* (1.4) −0.90 (−0.5) −0.38 (−1.2)
I1GDPGRO −6.23 (−0.9) −20.32** (−1.9) −213.85 (−0.5) 134.90 (0.9)
I1PRIMBS −17.20*** (−4.1) −19.07*** (−2.7) −14.54 (−0.2) 29.89 (0.4)
I1PUBINV 0.00 (0.5) 0.02 (0.5) 0.03 (0.3) 0.01 (0.7)
I1POLSTAB −0.11 (−0.5) 2.63 (0.9) −0.69** (−2.2) −1.88*** (−3.3)
I1LONG_US −2.6 (−1.0) −0.2 (−1.2) −1.37*** (−7.4) −0.85*** (−4.5)
I1CORRU 0.03 (0.3) −1.91 (−0.9) −0.46** (−1.7) −0.80** (−2.1)
δ ̂ 2 0.71*** (3.1) 0.30*** (2.7) 0.13*** (7.9) 0.13*** (4.7)
Usable obs. (n) 113 149 360 359
H0: One regime χ 9 2 = 290*** (0) χ 9 2 = 5,765*** (0) χ 10 2 = 1,349*** (0) χ 10 2 = 716*** (0)
H0: no serial corr. t61 = −0.6 (0.53) t124 = −0.6 (0.58) t246 = 3.7*** (0.00) t246 = 5.1*** (0.00)
H0: no break in 2008 F7,90 = 5.5*** (0.00) F6,130 = 2.7** (0.02) F4,330 = 5.9*** (0.00) F3,330 = 2.1* (0.09)
F11,86 = 6.2*** (0.00) F9,127 = 2.1** (0.03) F8,326 = 5.7*** (0.00) F7,326 = 3.1*** (0.00)

  1. (1) The routine NLLS of WinRATS 9.1 was used with the option lwindow = panel to obtain robust standard errors; (2) see also the Notes to Table 3.

Table 5:

GMM estimation of the log-linear model, Eq. (7), fixed country and time effects, Dep. variable: log of spread over Libor, M = Mean spread as threshold, R = Rating-based threshold.

Explanatory variables & tests Panel 1: 11 Eurozone countries Panel 2: 31 OECD countries
Regr. 9 (M) Regr. 10 (R) Regr. 11 (M) Regr. 12 (R)
Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat)
I −6.27*** (−13.8) −4.44*** (−18.0) −3.67*** (−16.2) −3.94*** (−16.8)
ILGAMMA −38.60*** (−7.3) −28.40*** (−5.0) −3.34 (−0.3) 3.60 (0.6)
IINFL 15.55*** (6.2) 14.16*** (3.6) 10.40** (1.7) 18.02*** (3.3)
IDEBTGRO 0.31 (0.8) 1.90* (1.6) 0.92 (0.4) 2.92** (1.9)
IUR 10.46*** (5.4) 9.16*** (6.0) 7.73*** (6.3) 6.91*** (3.8)
IGDPGRO −7.56*** (−4.5) −6.21** (−1.9) −10.69*** (−3.5) 0.23 (0.0)
IPRIMBS −0.34 (−0.5) 2.10 (1.0) −2.80 (−0.4) 1.12 (0.3)
IVIX 0.11*** (8.8) 0.02 (0.8) 0.02 (0.6) −0.03 (−0.7)
IPUBINV −0.00 (−0.6) −0.01*** (−2.7) −0.002** (−2.1) −0.002*** (−2.9)
IPOLSTAB 0.38 (1.1) −0.30 (−0.7) −0.41*** (−5.7) −0.36*** (−4.8)
ICORRU 0.14 (1.0) 0.24 (0.8) −0.12 (−0.5) −0.01 (−0.0)
IGOVEFFECT 0.10 (0.4) −0.01 (−0.0) −0.16 (−0.6) 0.01 (0.0)
I1 −5.95*** (−19.8) −3.61*** (−7.6) −2.67*** (−9.3) −3.11*** (−9.1)
I1LGAMMA −18.78*** (−5.1) −12.66 (−1.2) 30.61 (0.8) 22.37 (1.1)
I1INFL 6.53* (1.3) 11.40 (0.9) 4.36 (0.2) 2.18 (0.2)
I1DEBTGRO −1.35 (−1.3) 1.91 (1.0) 2.85* (1.6) −4.21 (−1.3)
I1UR 8.67*** (3.2) 4.79* (1.3) −2.85 (−0.3) 4.62* (1.4)
I1GDPGRO −5.54*** (−2.7) −6.81** (−1.7) −10.97 (−0.8) −14.28*** (−2.6)
I1PRIMBS −6.14*** (−3.6) −7.23*** (−2.7) 0.38 (0.0) 6.53 (0.9)
I1VIX 0.07*** (8.7) −0.02 (−0.6) 0.01 (0.2) −0.05 (−1.5)
I1PUBINV 0.00 (0.5) 0.01 (0.6) −0.001** (−1.7) −0.001** (−2.0)
I1POLSTAB −0.09 (−0.6) 0.29 (0.6) −0.01 (−0.1) −0.11 (−0.5)
I1CORRU −0.01 (−0.1) −0.46** (−1.7) −0.87*** (−5.7) −0.60*** (−3.6)
I1GOVEFFECT −0.13 (−0.9) −0.75** (−1.8) −0.01 (−0.1) 0.51 (0.4)
Usable obs. (n) 82 87 296 292
H0: one regime χ 6 2 = 39*** (0) χ 10 2 = 147*** (0) χ 9 2 = 4078*** (0) χ 9 2 = 2279*** (0)
p-value of J (d.f.) 0.09 (27) 0.58 (8) 0.19 (6) 0.74 (7)
H0: no serial corr. χ 1 2 = 0.7 (0.40) χ 1 2 = 0.6 (0.44) χ 1 2 = 0.0 (0.98) χ 1 2 = 0.3 (0.57)
H0: no break in 2008 χ 11 2 = 5.8 (0.89) χ 7 2 = 11.4 (0.12) χ 4 2 = 0.3 (0.99) χ 5 2 = 2.0 (0.85)
χ 13 2 = 10.6 (0.64) χ 13 2 = 121.8 (0.00) χ 6 2 = 0.0 (1.00) χ 7 2 = 0.0 (1.00)

  1. (1) In Regressions 9 and 10, the option “Jrobust = Distribution” is used, which produces a Jagannathan and Wang (1996) type weighting matrix, and the limiting distribution of the statistic J is non-standard, whereas in Regressions 11 and 12 no such option is used, so the limiting distribution of J is chi-squared; (2) the number in parentheses next to the p-value of J is the number of overidentifying restrictions; (3) see also the Notes to Table 3.

Table 6:

GMM estimation of the non-linear model, Eq. (10), fixed country-specific effects only, Dep. variable: Spread over Libor, M = Mean spread as threshold, R = Rating-based threshold.

Explanatory variables & tests Panel 1: 11 Eurozone countries Panel 2: 31 OECD countries
Regr. 13 (M) Regr. 14 (R) Regr. 15 (M) Regr. 16 (R)
Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat) Coeff. (t-stat)
I −10.46*** (−7.2) −9.85*** (−4.2) −7.16*** (−2.9) −2.95*** (−29.3)
IINFL 17.33** (2.3) 11.82** (1.8) 0.06 (0.0) 4.91 (0.9)
IDEBTGRO 1.87*** (5.0) 1.30*** (3.9) 1.87 (0.6) 1.09** (1.8)
IGDPGRO −8.24*** (−2.4) −9.82** (−2.6) −9.59** (−2.0) −8.48*** (−3.9)
IPRIMBS 0.49 (0.2) −2.34 (−1.1) −10.69*** (−3.1) −2.56** (−1.8)
IUR 13.39*** (5.4) 13.81*** (3.8) 0.27 (0.1) −6.90 (−1.2)
IVIX 0.24*** (3.7) 0.21*** (2.4) 0.19* (1.6) 0.02 (0.8)
ICORRU 3.77 (0.3) 0.28 (0.7) −0.33** (−1.7) −0.37*** (−5.8)
δ ̂ 1 0.44*** (3.6) 0.54*** (3.3) 0.34** (2.1) 0.79*** (4.4)
I1 −7.73 (−0.4) −3.44 (−1.1) −1.24 (−0.2) −1.70 (−0.3)
I1INFL −159.24 (−0.9) 84.92 (0.8) 8.96 (0.6) 18.43 (0.8)
I1DEBTGRO 3.75 (0.2) 7.16 (0.1) −0.78 (−0.5) 8.53 (1.1)
I1GDPGRO 2.00 (0.0) −104.13 (−0.6) −1.60 (−0.1) −67.17 (−0.4)
I1PRIMBS −196.48 (−0.2) −70.61 (−1.2) −1.40 (−0.2) −14.13 (−0.8)
I1UR −13.51 (−0.6) 5.50 (0.4) 0.13 (0.0) 6.71 (0.8)
I1VIX −0.07 (−0.5) −0.40 (−0.7) −0.1 (−0.4) 0.62 (0.4)
I1CORRU −2.80 (−0.2) 0.22 (0.1) 0.22 (0.7) 4.06 (0.2)
δ ̂ 2 0.05 (0.2) 0.10 (0.7) −1.93 (−0.8) 0.04 (0.2)
Usable obs. (n) 77 80 318 306
H0: one regime χ 7 2 = 12,942*** (0) χ 8 2 = 109,458*** (0) χ 6 2 = 1,619*** (0) χ 6 2 = 14,997*** (0)
p-value of J (d.f.) 0.10 (21) 0.39 (21) 0.12 (7) 0.15 (16)
H0: no serial corr. χ 1 2 = 0.0 (0.85) χ 1 2 = 1.0 (0.32) χ 1 2 = 1.6 (0.20) χ 1 2 = 0.01 (0.90)
H0: no break in 2008 χ 5 2 = 22.8*** (0.00) χ 5 2 = 39.6*** (0.00) χ 4 2 = 4.6 (0.33) χ 6 2 = 10.2 (0.12)
χ 6 2 = 20.2*** (0.00) χ 6 2 = 14.8** (0.02) χ 7 2 = 7.2 (0.41) χ 10 2 = 14.9 (0.14)

  1. (1) Here, the limiting distribution of the statistic J is chi-squared; (2) the Notes to Tables 35 apply here, too.

Before we discuss the results, the following comments are in order. First, the top (bottom) part of each table contains the coefficients for the high (low) spread regime. To get an idea of how “informed” each part is, note that for the Eurozone panel (respectively, for the OECD panel), when the definition M is used, on average, 33.8 (61.9) percent of the sample observations fit in the high-spread regime and 66.2 (38.1) percent in the low-spread regime; whereas in the case of the definition R, these percentages are 59.1 (41.8) and 40.9 (58.2), respectively.

Second, each regression has been estimated in accordance with the “general-to-specific” approach. That is, we begin with a long list of possible determinants of the spread; construct the interaction variables, e.g., IUR, I1UR, etc., so the initial list of variables doubles; and drop (normally, one at a time) the insignificant ones. We select the variables to be included in a regression based on economic as well as on empirical identification, i.e., correct signs and statistically significant coefficients at the 10 percent level or lower (Johansen and Juselius 1994, p. 8). This procedure is not straightforward, however. A variable that may appear to be statistically insignificant in one configuration of variables it may appear to be significant in a different configuration, where some previously “significant” variables may appear to be insignificant, perhaps because of multicollinearity.

Third, if a particular variable is dropped from a regression as insignificant, we report its coefficient and its t-statistic (obtained at a previous stage) in a shaded cell of Tables 36. Fourth, in the spirit of achieving both economic and empirical identification, when reporting the statistical significance of a coefficient, we view the alternative hypothesis as one-sided, in accordance with the expected sign of that coefficient. Thus, for example, in Regression 1 the sign of the coefficient of IREER_WB was expected to be negative, so the positive value of its t-statistic, t = 1.9, which would be viewed as significant at the 10-percent level if the alternative hypothesis was considered to be two-sided, it is now viewed as falling in the non-rejection region of the t-distribution, and IREER_WB is thus dropped. Fifth, a dash (−) in a cell implies that no estimate is available for that coefficient because of insufficient data on the corresponding variable (see Regressions 3 and 4 in Table 3).

We begin our discussion of the empirical results by considering Regression 1, which uses the data from Panel 1 and definition M for the “fair” spread to estimate Eq. (7) by least squares. Analogous comments apply to the other regressions, so they will be limited to comparisons amongst regressions only.

First, consider the signs and magnitudes of the coefficients of Regression 1. The estimate of β0 is −4.51, which does not seem to be unreasonable, since the sample mean of the logarithm of the spread in Panel 1 is −4.2648 (see Table 1). The signs of the other coefficients are as expected: (1) negative for LGAMMA, since, by definition, the risk-free rate (r*) negatively influences the spread (s = rr*); (2) positive for the inflation rate (the Fisher effect); (3) positive for the rate of growth of the debt-to-GDP ratio, higher values of which erode the country’s creditworthiness and raise the spread; (4) positive for the unemployment rate, as recessions reduce creditworthiness; (5) negative for the growth rate of real GDP (by the same token); (6) positive for the uncertainty index, as uncertainty raises the risk of default; (7) negative for the primary budget surplus, as a higher primary surplus improves a country’s ability to service its debt; (8) negative for the real exchange rate, as a real depreciation of the home currency raises the country’s competitiveness in international markets, thus strengthening its economy; (9) negative for private investment, which also strengthens the economy; and (10) negative for the quality-of-institutions indicators, such as the index for political stability, higher values of which indicate better quality of institutions, hence higher credibility and lower spread.

Note that the coefficients are semi-elasticities. Thus, for example, in Regression 1, a ceteris-paribus increase in the inflation rate (INFL) by one percentage point, e.g., from 0.04 to 0.05 is expected to raise the spread by 12.18 percent in the high-spread regime, e.g., from 0.03 to 0.033654, but leave it intact in the low-spread regime, as the coefficient of I1NFL, 1.34 (t = 0.30), is taken to be zero, hence I1NFL was dropped from the regression. As another example, since the coefficients of IVIX and I1VIX in Regression 1 are 0.04 and 0.01, respectively, a ceteris-paribus increase in the uncertainty index by one unit, say, from 20 to 21, is expected to raise the spread by 4 percent in the high-spread regime, e.g., from 0.03 to 0.0312, and by 1 percent in the low-spread regime, e.g., from 0.03 to 0.0303.

Second, the evidence supports the two-regime “threshold-value hypothesis,” as in most cases the coefficient of an explanatory variable is larger (in absolute value) in the high-spread than in the low-spread regime, suggesting that its impact on the spread is larger in the high-spread regime. In each regression, we formally test the one-regime hypothesis, that the corresponding coefficients in the low-spread and in the high-spread regime are equal. In the context of Eq. (7), this amounts to testing the following restrictions on the coefficients: β0 = β0′, β j = β j ′, α = α′, j = 1, …, k. Note, if a coefficient, say, β j ′, turns out to be statistically insignificant, we set β j ′ = 0, so the restriction tested in this case is β j = 0. The one-regime hypothesis is strongly rejected in every regression of Tables 36, as the p-value of the test is always zero. In addition, in 94 out of 128 pairs of coefficients (β0, β0′), (β j , β j ′), and (α, α′) reported as statistically significant in Regressions 1–16, the unprimed coefficient is larger than its counterpart primed one (in absolute value). This is consistent with the hypothesis that the demand for loans is less elastic at high than at low spreads (see Section 1). Using a quantile regression, Gomez-Bengoechea and Arahuetes (2019) find a similar result, namely the impact of some macroeconomic variables (e.g., public debt and the unemployment rate) on the spread is larger at higher than at lower spreads.

Third, we now turn to the evaluation of Regression 1 using several diagnostic tests. Based on the RESET, which essentially tests for nonlinearities, the results can be considered reliable, as the p-value of the squared fitted value included as an additional regressor in (7) is 0.50, so the null hypothesis of a log-linear model cannot be rejected. In addition, the coefficient of determination (R2) is high (0.95). Regression 1 also passes the serial correlation test (p-value = 0.40). This test is based on a Gauss-Newton regression (GNR) described in MacKinnon (1992, Section 2, pp. 110–112) and in Davidson and MacKinnon (1993, pp. 357–358). Regression 1 fails, however, the structural stability test. The null hypothesis in this case is that the regression coefficients remained stable after 2008, when the financial and economic crisis took effect.[7] This test is also based on a GNR; see MacKinnon (1992, Section 3, p. 114) and Davidson and MacKinnon (1993, pp. 377–381). Finally, there is evidence that the endogeneity problem discussed in Sections 2 and 3 is serious. In particular, the Durbin-Wu-Hausman (DWH) test strongly rejects the null hypothesis that the least squares estimates of Regression 1 are consistent (Davidson and MacKinnon 1993, pp. 237–242). The same is true for Regressions 2–4. Note that the DWH test is implemented in Regressions 1–4 by employing the instrumental variables (IVs) used in the estimation of Regressions 9–12, respectively; and since it strongly rejects the null hypothesis in Regressions 1–4, we consider the latter unreliable.

Next, assuming no time effects, we estimate the nonlinear model, Eq. (10), by nonlinear least squares (NLLS) with robust standard errors and report the results in Table 4. Qualitatively, these results do not differ from those of the log-linear model. In particular, note that the evidence supports again the “threshold-value hypothesis.” It also supports the hypothesis of partial adjustment, as the estimates of δ1 and δ2 are between zero and one and are highly significant. It is worth noting that in Regressions 6–8 the estimate of δ1 is larger than that of δ2, which suggests that adjustment occurs faster in the high-spread regime, in that government-bond yield spreads are affected more quickly by information on government creditworthiness and financial health in the high-spread regime than in the low-spread regime. Note that Regressions 5–6 pass the serial-correlation test, but are subject to the endogeneity problem discussed earlier. In addition, all of the Regressions of Table 4 fail the structural stability test at the 5-percent level. It should be noted, however, that if the whole set of regression coefficients (including those of the dummies for the fixed effects) is tested for stability after 2008, only Regressions 15 and 16 pass the test, perhaps because they are parsimonious.[8] Thus, in Regressions 1–14, we can only opt for testing the stability of only a subset of coefficients, especially those of the macroeconomic variables.

As a reaction to the endogeneity problem addressed earlier, we now estimate Eqs. (7) and (10) by GMM with robust standard errors and report the results in Tables 5 and 6. In Regressions 9 and 10, the efficient GMM estimator, which uses the “optimal” weighting matrix suggested by Hansen and Singleton (1982), fails, apparently because of the relatively small sample size (Hayashi 2000, p. 215), so we use a weighting matrix in line with Jagannathan and Wang (1996). As a consequence, the limiting distribution of the well-known J statistic for testing the overidentifying restrictions is not chi-squared, but non-standard (RATS User’s Guide 2014, p. 140). In Regressions 11 and 12, however, where the sample size is larger, the “optimal” weighting matrix works well.

Note, first, that in an effort to produce statistically significant estimates, we use a different instrument set in each of the eight regressions of Tables 5 and 6. Second, the sizes (but not the signs) of the estimated coefficients differ across models and data. Third, in most cases, the “threshold-value hypothesis” is supported again, most sharply by Regressions 9 and 13–16, especially by the last four, where all coefficients in the low-spread regime are not statistically different from zero (see Table 6). Figure 2 provides a possible explanation for this phenomenon: if the supply curve of loans is almost perfectly elastic at low interest rates, only a negligible increase in the interest rate is required to restore equilibrium after a large increase in demand (see Section 1). Fourth, on the basis of the J statistic, these results can be considered reliable at the 5-percent level. Fifth, all of the regressions of Tables 5 and 6 pass the serial correlation test. Sixth, Regressions 9–12 and 15–16 pass the structural stability test, at least for the subset of coefficients of the macroeconomic variables. Thus, since these regressions are estimated by GMM and are thus not expected to suffer from the endogeneity problem discussed in Section 2, they can be considered fairly reliable, and hence usable. Finally, note that in Regressions 15 and 16, which are estimated with data from the wider OECD area, the coefficient of the primary surplus is consistent with our “threshold-value hypothesis,” since it has the expected sign and is significant at the high-spread regime, but is zero at the low-spread regime. This result, which also emerges in Regression 4, is at variance, however, with that obtained in seven of our regressions (1–3, 5–6, and 9–10), six of which use the data from the 11 Eurozone countries. In these six regressions (1–2, 5–6, and 9–10), contrary to our intuition, the primary surplus, on which the well-known financial institutions focus their attention, matters only in the low-spread regime. Taken at face value, this finding means that, for the Eurozone countries that suffer from high spreads, the insistence on achieving primary surpluses in order to reduce the debt-to-GDP ratio may not be an appropriate policy, as pursuing such targets may further deteriorate the economy and increase the probability of default. Cantor and Packer (1996) report a similar result.

5 Summary and conclusions

Broadly speaking, our results support the following conclusions. First, our estimates support the “threshold-value hypothesis,” as in most cases the coefficient of a given explanatory variable is larger (in absolute value) in the high-spread regime than in the low-spread regime. In particular, based on our preferred regressions (9, 15, and 16), we conclude that an improvement in the macroeconomic environment, e.g., lower unemployment, lower inflation, lower growth of the debt-to-GDP ratio, less macroeconomic uncertainty, and higher growth of real GDP reduce the spread facing a country (by enhancing its creditworthiness) to a greater extent in high-spread situations than in low-spread situations. Regressions 15 and 16 take this to the extreme, since they imply that the spread reacts noticeably to changes in the above variables, as well as in the primary surplus, only in the high-spread regime. In terms of Figures 1 and 2, this can happen if the demand for and supply of loans have relatively low elasticities at high rates, and the supply curve is almost perfectly elastic at low rates. This scenario may not be unreasonable in long-run competitive equilibrium, where at relatively low levels of output the constant-cost industry assumption holds approximately, but at higher levels of output costs are increasing, thus giving rise to an upward-sloping supply curve.

Second, an improvement in the quality of institutions (e.g., less corruption) lowers the spread to a greater extent in high-spread situations than in low-spread situations. We interpret this to imply that quality of institutions is particularly important for countries facing multidimensional problems (e.g., high unemployment), which put them among the high-spread countries.[9]

Third, the signs (but not the sizes) of the coefficients are fairly robust to substantial changes in the following: (1) the functional form of the model (log-linear vs. non-linear); (2) the sample (Panel 1 vs. Panel 2); (3) the method of estimation (NLLS vs. GMM); and (4) the definition of the “fair” value of the spread (mean spread vs. rating-based spread).

Fourth, the evidence presented in Tables 36 suggests that the economic and financial crisis of 2008 caused at least some of the regression coefficients to change.

Fifth, because of the endogeneity problem discussed in Sections 2 and 3, we consider the GMM estimates to be more reliable than the least-squares ones.

Sixth, the definition of the “fair” spread in year t as the average spread facing all the countries in the panel in that year seems to be more consistent with the rest of the paper, because it introduces less endogeneity and is more supportive of our “threshold-value hypothesis,” as Regression 9 demonstrates.

Finally, based on Regressions 13–16, especially the last two, the only two regressions that remained stable as a whole after 2008; and also recalling our caution that the log transformation might not be innocuous after all (see Section 2); we tend to conclude that for the problem at hand the nonlinear model that embodies the partial adjustment hypothesis may be considered preferable to the log-linear one. Nevertheless, the log-linear Regression 9 is also one of our preferred regressions, as it does not seem to suffer from any problems, and is supportive of our “threshold-value hypothesis.”

Corresponding author: Dimitris Hatzinikolaou, Department of Economics, University of Ioannina University Campus, 45110 Ioannina, Greece; and Hellenic Open University, 18 Aristotelous, Patras 263 35, Greece, E-mail:


The paper is an extensively revised version of the doctoral dissertation of the second author, completed in 2017 at the University of Ioannina, Department of Economics. We thank Professors Angelos A. Antzoulatos and Athanasios G. Noulas for their valuable comments and suggestions. We are also grateful to two anonymous referees of this Journal for their constructive comments, which improved the paper significantly. The usual disclaimer applies.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


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Supplementary Material

The online version of this article offers supplementary material (

Received: 2020-01-20
Revised: 2022-01-20
Accepted: 2022-05-09
Published Online: 2022-06-07

© 2022 Dimitris Hatzinikolaou and Georgios Sarigiannidis published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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