Estimation Method

If the threshold were known, the model could be estimated by ordinary least squares (OLS). Since π^* is unknown, it has to be estimated along with the other regression parameters. The appropriate estimation method in this case is nonlinear least squares (NLLS). Furthermore, since π^* enters the regression in a nonlinear and non-differentiable manner, conventional gradient search techniques to implement NLLS are inappropriate. Instead, estimation has been carried out with a method called conditional least squares, which can be described as follows. For any π^*, the model is estimated by OLS, yielding the sum of squared residuals as a function of π^*. The least squares estimate of π^* is found by selecting the value of π^* which minimizes the sum of squared residuals. Stacking the observation in vectors yields the following compact notation for equation (1):

where ${\beta }_{\pi }={\left({\mu }_i{\mu }_t{\gamma }_1{\gamma }_2{\theta }^{\prime }\right)}^{\prime }$ is the vector of parameters and X is the corresponding matrix of observations on the explanatory variables. Note that the coefficient vector β is indexed by n to show its dependence on the threshold level of inflation, the range of which is given by π

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Inference

It is important to determine whether the threshold effect is statistically significant. In equation (1), to test for no threshold effects amounts simply to testing the null hypothesis H0: γ₁ = γ₂. Under the null hypothesis, the threshold π^* is not identified, so classical tests, such as the t-test, have nonstandard distributions. Hansen (1996, 1999) suggests a bootstrap method to simulate the asymptotic distribution of the following likelihood ratio test of H0:

where S₀, and S₁ are the residual sum of squares under H0: γ₁ = γ₂, and H1: γ₁ ≠ γ₁, respectively; and ${\hat{\sigma }}^2$ the residual variance under H₁. In other words, S₀ and S₁ are the residual sum of squares for equation (1) without and with threshold effects, respectively. The asymptotic distribution of LR₀ is nonstandard and strictly dominates the χ² distribution. The distribution of LR₀ depends in general on the moments of the sample; thus critical values cannot be tabulated. Hansen (1999) shows how to bootstrap the distribution of LR₀.

An interesting question is whether an inflation threshold, for example, of 10 percent is significantly different from a threshold of 8 percent or 15 percent. In other words, can the concept of confidence intervals be generalized to threshold estimates? Chan and Tsay (1998) show that in the case of a continuous threshold model studied here, the asymptotic distribution of all parameters, including the threshold level, have a normal distribution.¹¹ More precisely, define $\mathrm{\Phi }=({\mu }_i{\mu }_t{\gamma }_1{\gamma }_2{\theta }^{\prime },{\pi }^{*})$ as the set of all parameters, including the threshold level. Chan and Tsay (1998) show that the NLLS estimate Φ of Φ (described above) is asymptotically normally distributed:¹²

where $U=E\left(H_{it}{H}_{it}^{\prime }\right),$ $V=E\left(e_{it}^2{H}_{it}{H}_{it}^{\prime }\right),$ $H_{it}=(-{\tilde{x}}_{it},{\gamma }_1(1-{d_{it}}^{{\Pi }^{*}})+{\gamma }_2{d_{it}}^{{\Pi }^{*}}),$${\tilde{X}}_{it}$ is the vector of all right-hand-side variables in equation (1), and NT is the total number of observations. An estimate of U and V are given by $\hat{U}=\underset{i=1}{\overset{N}{\mathrm{\Sigma }}}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\hat{H}}_{it}{\hat{H}}_{it}^{\prime }/\left(NT\right)$ and $\hat{V}=\underset{i=1}{\overset{N}{\mathrm{\Sigma }}}\underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\hat{e}}_{it}^2{\hat{H}}_{it}{\hat{H}}_{it}^{\prime }/\left(NT\right)$with ${\hat{H}}_{it}=(-{\tilde{X}}_{it},{\hat{\gamma }}_1(1-{d_{it}}^{{\pi }^{*}})+{\hat{\gamma }}_2{d_{it}}^{{\pi }^{*}})\mathrm{.}$

Test for Existence of Threshold Effects

The first step is to test for the existence of a threshold effect in the relationship between real GDP growth and inflation using the likelihood ratio, LR₀, discussed above. This involves estimating equation (1) and computing the residual sum of squares (RSS) for threshold levels of inflation ranging from π

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Table 1.

Test Results of Threshold Effects

Note: The second column gives the range over which the search for the threshold effect is conducted, the third column gives the threshold estimate in percent, the column LR₀ gives the observed value of the likelihood ratio, the fifth column gives the critical values, and the last column gives the corresponding significance level, both computed using the bootstrap distributions (corresponding to the three samples) of LR0. For a more detailed discussion on the computation of the bootstrap distribution of LR₀, see Hansen (1999).

Table 1.

Test Results of Threshold Effects

	Search Range	Threshold		Critical	Significance
Sample	for Thresholds	Estimate (%)	LR 0	Values	Levels
All Countries	{1, 2, 3…..100}	11	10.59	7.47	0.001
Industrial Countries	[1, 2, 3…..30]	1	8.80	6.63	0.005
developing Countries	{1, 2, 3…..100]	11	10.8"	6.21	0.000

Note: The second column gives the range over which the search for the threshold effect is conducted, the third column gives the threshold estimate in percent, the column LR₀ gives the observed value of the likelihood ratio, the fifth column gives the critical values, and the last column gives the corresponding significance level, both computed using the bootstrap distributions (corresponding to the three samples) of LR0. For a more detailed discussion on the computation of the bootstrap distribution of LR₀, see Hansen (1999).

View Table

Table 1.

Test Results of Threshold Effects

	Search Range	Threshold		Critical	Significance
Sample	for Thresholds	Estimate (%)	LR 0	Values	Levels
All Countries	{1, 2, 3…..100}	11	10.59	7.47	0.001
Industrial Countries	[1, 2, 3…..30]	1	8.80	6.63	0.005
developing Countries	{1, 2, 3…..100]	11	10.8"	6.21	0.000

Note: The second column gives the range over which the search for the threshold effect is conducted, the third column gives the threshold estimate in percent, the column LR₀ gives the observed value of the likelihood ratio, the fifth column gives the critical values, and the last column gives the corresponding significance level, both computed using the bootstrap distributions (corresponding to the three samples) of LR0. For a more detailed discussion on the computation of the bootstrap distribution of LR₀, see Hansen (1999).

The second column gives the range over which the search for the threshold effect is conducted. For the full sample, π

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Residual Sum of Squares as a Function of the Threshold Level

(Five-Year Average)

Citation: IMF Staff Papers 2001, 003; 10.5089/9781451974614.024.A001

Note: Figure 3 shows the residual sum of squares (RSS) from equation (1) as a function of the threshold level of inflation for the three samples. The minimum of the RSS sequence determines the threshold estimate, which occurs at 1 percent for industrial countries, and 11 percent for developing countries and the full sample.

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Estimation Results

Table 2 provides the estimation results of equation (1) for the three samples. Fixed effects and time dummies have been included (but not reported) to control for cross–country heterogeneity and time effects. For the full sample, for which the threshold estimate is 11 percent, all coefficients have the right sign and are statistically significant at the 1 percent level. Recall that the existence of a threshold effect cannot be inferred simply from a classical test of equality between γ₁ and γ₂ as the distribution of the t-statistic for this variable is highly nonstandard under the null hypothesis of no threshold effect. This is why the null hypothesis has been tested using the bootstrap distribution of the likelihood ratio LR₀(π). However, the distribution of the t-values of all explanatory variables retain their usual distribution under the alternative hypothesis of a threshold effect. Furthermore, Chan and Tsay (1998) show that the asymptotic distribution of all coefficients, including the threshold, is multivariate normal with a variance-covariance matrix given by equation (6).

Table 2.

NLLS With Fixed Effects

(Five-Year Average)

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

Table 2.

NLLS With Fixed Effects

(Five-Year Average)

	Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{{\pi }^{*}})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00049	0.05991	0.00109
	(-0.66)	(2.53)a	(1.33)
$d^{{\pi }^{*}}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.00895	-0,00643	-0.00895
	(-4.70)a	(-4.23)a	(-4.42)a
lyo	-0.02506	-0.03634	-0.02551
	(-13.20)a	(-15.58)a	(-11.08)a
igdp	0.15090	0.10640	0.15910
	(5.01)a	(3.47)a	(4.96)a
dlog(pop)	0.04947	-O.01557	0.05095
	(2,33)a	(-0.23)	(2.33)a
σ tot	-0.00020	-0.00031	-0.00019
	(-2.56)a	(-1.17)	(-2.33)4
Threshold {%)	11	1	11
	(64.42)a	(9.10)a	(58.59)a
NxT	905	165	740
R2	0.43	0.80	0.39

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

View Table

Table 2.

NLLS With Fixed Effects

(Five-Year Average)

	Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{{\pi }^{*}})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00049	0.05991	0.00109
	(-0.66)	(2.53)a	(1.33)
$d^{{\pi }^{*}}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.00895	-0,00643	-0.00895
	(-4.70)a	(-4.23)a	(-4.42)a
lyo	-0.02506	-0.03634	-0.02551
	(-13.20)a	(-15.58)a	(-11.08)a
igdp	0.15090	0.10640	0.15910
	(5.01)a	(3.47)a	(4.96)a
dlog(pop)	0.04947	-O.01557	0.05095
	(2,33)a	(-0.23)	(2.33)a
σ tot	-0.00020	-0.00031	-0.00019
	(-2.56)a	(-1.17)	(-2.33)4
Threshold {%)	11	1	11
	(64.42)a	(9.10)a	(58.59)a
NxT	905	165	740
R2	0.43	0.80	0.39

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

In the previous sub-section, we established the existence of a threshold for all three samples; the next important question is how precise are these estimates? This requires the computation of the confidence region around the threshold estimate. While the existence of threshold effects in the relationship between inflation and growth is well accepted, the precise level of the inflation threshold is still subject to debate. Indeed, as discussed earlier, based on existing studies, the range could be between 2.5 percent and 40 percent. If the confidence region shows that the threshold estimate is not significantly different from a large number of other potential threshold levels, that would imply that there is substantial uncertainty about the threshold level. Interestingly, the confidence intervals here are very tight, which implies that the thresholds are precisely estimated. Indeed, the 95 percent confidence intervals for the whole sample, industrial countries, and developing countries, are [10.66, 11.34], [0.89, 1.11], and [10.62, 11.38], respectively.

Two basic conclusions can be drawn from this set of statistical tests. First, the threshold is around an inflation rate of 1 percent for industrial economies and 11 percent for developing countries. Second, these threshold estimates are very precise. One needs to ask why the threshold level for developing countries is higher than the threshold level for industrial countries. There are at least two possible conjectures that we can make. First, the long history of inflation in many developing countries led them to adopt widespread indexation systems to negate, at least partially, the adverse effects of inflation. Once in place, these indexation mechanisms make it possible for governments in these countries to run higher rates of inflation without experiencing adverse growth effects (because relative prices do not change that much). Second, to the extent that inflation is viewed as a tax on financial intermediation, governments, faced with a target level of expenditure will, in the absence of conventional taxes, levy the inflation tax. Accordingly, the differential threshold levels for the effects of inflation on growth for industrial and developing countries could reflect the higher level of conventional taxation in the former than in the latter. Thus, while relatively small increases in inflation in industrial countries adversely affect investment (by raising the effective cost of capital goods), productivity, and growth, in developing countries, with relatively low levels of conventional taxes, a larger inflation tax is required to have the same growth-inhibiting effects.¹⁶

While inflation below its threshold level has no significant effect on growth, inflation rates above the threshold level have a significant negative effect on growth for the whole sample. Dividing the sample into industrial and developing countries yields some interesting insights. First, both groups show a positive relationship between growth and inflation below their respective threshold levels (although it is statistically significant only for industrial countries for which the threshold level is at 1 percent), and a significant and a more powerful negative relationship for inflation rates above the threshold. As expected, investment as a share of GDP and population growth have a positive and significant effect on growth (except for industrial countries for which population growth is statistically insignificant). On average, an increase in the investment- GDP ratio of 5 percentage points will boost real GDP growth by 0.80 percentage points for developing countries and by 0.53 percentage points for industrial countries. In the empirical growth literature, the log of the initial GDP per capita (ly₀) has been generally included in growth regressions to test conditional convergence. Conditional convergence holds if the coefficient on ly₀ is negative.¹⁷ Thus, convergence occurs for all samples. The rate of convergence among industrial countries is faster than for developing countries, corroborating the results of previous studies, which find that conditional convergence is stronger among industrial countries.¹⁸

The first three panels of Table 3 illustrate the regression results reported in Table 2 for the full sample, for industrial countries, and for developing countries, respectively. The three panels show the effect on growth of gradually increasing inflation for a hypothetical economy with an initial inflation rate of 3 percent.¹⁹ The maximum growth that a developing country, with an initial inflation rate of 3 percent, can gain through further inflation is 0.14 percentage points (by moving from an annual inflation rate of 3 percent to 11 percent). This magnitude very likely overestimates the positive effect of inflation as investment over GDP (igdp) was held constant while moving inflation from 3 to 11 percent. However, Fischer (1993) has shown that inflation also has a negative and significant indirect effect on growth through its effect on investment. This indirect effect is not taken into account here. From our results, the positive effect rapidly changes into a negative one as inflation increases above the threshold. For example, an increase in inflation from 3 to 40 percent will reduce growth by 1.01 percentage points in developing countries and by 1.66 percentage points in industrial countries. The effect of inflation on growth for any pair of inflation rates in the first column is simply equal to the difference between their growth effects. For example, reducing a developing country’s annual inflation rate from 60 percent to 15 percent will increase its GDP growth by 1.24 percentage points. The log transformation implies that the effect on growth will be identical for an economy that moves from a 3 percent inflation rate to 6 percent and an economy that increases its inflation rate from 4 percent to 8 percent. This is because, in both cases, the inflation rate is doubled. Of course, this property holds only for inflation changes that do not induce a crossing of the threshold.

Table 3.

Numerical Illustration of the Effects of Inflation on Growth

(In percent)

Note: This table shows the effect on growth of gradually increasing inflation from an initial inflation rate (π₀) of 3 percent to 60 percent, using estimates of the fixed-effects model with yearly and five-year-average data. For example, increasing inflation from 3 percent to 25 percent entails a loss in growth of 1.17 percent using the full sample estimates with yearly data. Shaded areas indicate a crossing of a threshold.

Table 3.

Numerical Illustration of the Effects of Inflation on Growth

(In percent)

πo= 3 percent
		Five-Year-Average				Yearly Data
	All	Industrial	Developing		All	Industrial	Developing
Threshold π	11%	1%	11%		9%	3%	12%
4	0.01	-0.18	0.03		0.02	-0.27	0.01
5	0.02	-0.33	0.06		0.03	-0.47	0.02
6	0.03	-0.45	0.08		0.04	-0.64	0.03
9	0.05	-0.71	0.12		0.06	-1.01	0.05
11	0.06	-0.84	0.14		-0.18	-1.20	0.06
15	-0.21	-1.03		-0.14	-0.54	-1.49	-0.24
20	-0.47	-1.22		-0.39	-0.88	-1.75	-0.63
25	-0.67	-1.36		-0.59	-1.15	-1.96	-0.93
30	-0.83	-1.48		-0.76	-1.36	-2.13	-1.18
40	-1.09	-1.66		-1.01	-1.70	-2.39	-1.56
60	-1.45	-1.93		-1.38	-2.18	-2.77	-2.11

Note: This table shows the effect on growth of gradually increasing inflation from an initial inflation rate (π₀) of 3 percent to 60 percent, using estimates of the fixed-effects model with yearly and five-year-average data. For example, increasing inflation from 3 percent to 25 percent entails a loss in growth of 1.17 percent using the full sample estimates with yearly data. Shaded areas indicate a crossing of a threshold.

View Table

Table 3.

Numerical Illustration of the Effects of Inflation on Growth

(In percent)

πo= 3 percent
		Five-Year-Average				Yearly Data
	All	Industrial	Developing		All	Industrial	Developing
Threshold π	11%	1%	11%		9%	3%	12%
4	0.01	-0.18	0.03		0.02	-0.27	0.01
5	0.02	-0.33	0.06		0.03	-0.47	0.02
6	0.03	-0.45	0.08		0.04	-0.64	0.03
9	0.05	-0.71	0.12		0.06	-1.01	0.05
11	0.06	-0.84	0.14		-0.18	-1.20	0.06
15	-0.21	-1.03		-0.14	-0.54	-1.49	-0.24
20	-0.47	-1.22		-0.39	-0.88	-1.75	-0.63
25	-0.67	-1.36		-0.59	-1.15	-1.96	-0.93
30	-0.83	-1.48		-0.76	-1.36	-2.13	-1.18
40	-1.09	-1.66		-1.01	-1.70	-2.39	-1.56
60	-1.45	-1.93		-1.38	-2.18	-2.77	-2.11

Note: This table shows the effect on growth of gradually increasing inflation from an initial inflation rate (π₀) of 3 percent to 60 percent, using estimates of the fixed-effects model with yearly and five-year-average data. For example, increasing inflation from 3 percent to 25 percent entails a loss in growth of 1.17 percent using the full sample estimates with yearly data. Shaded areas indicate a crossing of a threshold.

Sensitivity to Fixed Effects

Since panel estimation can be quite sensitive to the use of fixed effects, equation (1) has also been estimated without fixed effects. Tables 2 and 4 show similar results. In particular, the estimates of threshold levels are identical. However, omitting fixed effects weakens the negative effect of inflation on growth for developing countries above the threshold level of inflation, and lowers the rate of convergence among countries.

Sensitivity to High-Inflation Observations

Bruno and Easterly (1998) and Easterly (1996) have argued that the negative relationship between inflation and growth holds only for high-inflationary economies. They show that excluding observations with annual inflation rates of 40 percent or more weakens the negative relationship between inflation and growth. Their methodology differs from ours in that theirs is not based on regression analysis but on mean comparisons before, during, and after inflation crises (defined as inflation episodes above 40 percent). To test their hypothesis within our framework, equation (1) was re-estimated with five-year-averaged data excluding observations with inflation rates higher than 40 percent. The results are presented in Table 5.

Table 4.

NLLS Without Fixed Effects

(Five-Year Average)

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

Table 4.

NLLS Without Fixed Effects

(Five-Year Average)

	Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{{\pi }^{*}})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00061	0.05667	0.00074
	(0.98)	(2.85)a	(1.11)
$d^{{\pi }^{*}}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.00574	-0.00737	-O.00586
	(-2.71)a	(-5.00)a	(-2.65)a
ly 0	-0.00262	-0.02582	-0,00228
	(-7.52)a	(-67.76)a	(-4.47)a
igdp	0.11580	0.09150	0.10810
	(7.78)a	(10.93)a	(6.10)a
dlog(pop)	0.03873	0.27128	0.03234
	(3.06)a	(8.15)a	(2.08)b
σ fot	-0,00021	-0.00104	-0.00019
	(-2.61)a	(-4.54)4	(-2.24)b
Threshold estimate (%)	11	1	1 1
	(25.42)a	(7.40)a	(23.97)a
NxT	905	165	740
R2	0.21	0.70	0.19;

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

View Table

Table 4.

NLLS Without Fixed Effects

(Five-Year Average)

	Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{{\pi }^{*}})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00061	0.05667	0.00074
	(0.98)	(2.85)a	(1.11)
$d^{{\pi }^{*}}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.00574	-0.00737	-O.00586
	(-2.71)a	(-5.00)a	(-2.65)a
ly 0	-0.00262	-0.02582	-0,00228
	(-7.52)a	(-67.76)a	(-4.47)a
igdp	0.11580	0.09150	0.10810
	(7.78)a	(10.93)a	(6.10)a
dlog(pop)	0.03873	0.27128	0.03234
	(3.06)a	(8.15)a	(2.08)b
σ fot	-0,00021	-0.00104	-0.00019
	(-2.61)a	(-4.54)4	(-2.24)b
Threshold estimate (%)	11	1	1 1
	(25.42)a	(7.40)a	(23.97)a
NxT	905	165	740
R2	0.21	0.70	0.19;

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation, π; the log of initial income, ly₀; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ(tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

Table 5.

NLLS With Fixed Effects

(Five-Year Average)

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation,π; the log of initial income, ly_o’, gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ (tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

Table 5.

NLLS With Fixed Effects

(Five-Year Average)

Excluding Observations With Inflation Greater Than 40 Percent Dependent Variable: dlog(gdp)
Independent Variables	All	industrial	Developing
$(1-d^{\pi *})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00101	0.06227	0.00166
	(1.34)	(2.77)a	(l.85)c
$d^{\pi *}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.01983	-0.00709	-0.02067
	(-9.04)a	(-4.10)a	(-9.02)a
ly 0	-0.02286	-0,03461	-0.02299
	(-9.00)a	(-13.91)8	(-6.89)a
igdp	0.15100	0.09860	O.161OO
	(4.71)a	(2.83)a	(4.77)a
dlog(pop)	0.06509	-0.00132	0.06595
	(3.06)a	(-0.02)	(3.01)a
σ tot	-0.00027	-0.00040	-0.00026
	(-2.43)a	(-0.98	(-2.25)b
Threshold estimate (%)	12	1	12
	(130.21)a	(9.94)a	(123.1l)a
NxT	838	160	678
R2	0.43	0.80	0.39

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation,π; the log of initial income, ly_o’, gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ (tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

View Table

Table 5.

NLLS With Fixed Effects

(Five-Year Average)

Excluding Observations With Inflation Greater Than 40 Percent Dependent Variable: dlog(gdp)
Independent Variables	All	industrial	Developing
$(1-d^{\pi *})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00101	0.06227	0.00166
	(1.34)	(2.77)a	(l.85)c
$d^{\pi *}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.01983	-0.00709	-0.02067
	(-9.04)a	(-4.10)a	(-9.02)a
ly 0	-0.02286	-0,03461	-0.02299
	(-9.00)a	(-13.91)8	(-6.89)a
igdp	0.15100	0.09860	O.161OO
	(4.71)a	(2.83)a	(4.77)a
dlog(pop)	0.06509	-0.00132	0.06595
	(3.06)a	(-0.02)	(3.01)a
σ tot	-0.00027	-0.00040	-0.00026
	(-2.43)a	(-0.98	(-2.25)b
Threshold estimate (%)	12	1	12
	(130.21)a	(9.94)a	(123.1l)a
NxT	838	160	678
R2	0.43	0.80	0.39

Note: The panel has 8 observations (T), that is five-year averages over 1960-98, for 140 countries (N). The variables are inflation,π; the log of initial income, ly_o’, gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the standard deviation of terms of trade, σ (tot). The dummy variable d^π* takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated time dummies and country-specific effects are not reported.

The results turn out to be very close to the estimates with the full sample (given in Table 2). In fact, the threshold estimates without high inflation observations for developing countries are almost identical to the estimates obtained with all the data.²⁰

Sensitivity to the Location of the Threshold

Figure 4 shows the sensitivity of the effect of inflation on growth when the threshold level varies from 1 percent to 50 percent. The three panels (corresponding to the three samples) depict the effect of inflation on growth for economies with an inflation rate below the threshold level (solid line) and for economies with inflation rates above the threshold level (dotted line). These effects are given by the coefficients γ₁ and γ₂ in equation (1). The vertical line indicates the threshold estimate. The following points emerge from Figure 4:

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Sensitivity of the Effect of Inflation on Growth to the Threshold Level

(Five-Year Average)

Citation: IMF Staff Papers 2001, 003; 10.5089/9781451974614.024.A001

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• the high and low-inflation effects are most sensitive to the location of the threshold over the 1 to 20 percent range;

• the positive effect of inflation on growth is only present for inflation rates lower than 5 percent for industrial countries and 18 percent for developing countries;

• for developing countries, the inflation effect on growth, which is negative over the whole range, strengthens as the threshold increases, which implies a worsening of the negative effect of inflation on growth as inflation increases; and

• for industrial countries, the inflation effect, while remaining negative over the entire range, first weakens (in absolute value) as the inflation threshold increases, reaches a minimum around a threshold of 15 percent, and strengthens thereafter.

Sensitivity to Data Frequency

The estimation and inference in the previous section were based on five-year averages of the data. This procedure has become common practice in empirical growth literature and aims at filtering out business cycle fluctuations and allowing the focus to be on the medium- and long-term trends in the data. Estimation of equation (1) has also been carried out with annual data in order to examine two issues. First, it is interesting to analyze how data frequency changes the location and the magnitude of the threshold effect and the parameter estimates of equation (1). Second, while noisier, annual data provide more degrees of freedom, especially at the tails of the distribution for inflation. In particular, the inflation threshold for industrial countries was estimated at 1 percent, which was the lower bound of the grid search for threshold effects. The question raised earlier was whether the threshold was at 1 percent or at less than 1 percent. With the five-year averages, there were not enough observations with inflation at less than 1 percent, whereas annual data provide enough low-inflation observations to answer the question.

Table 6 gives the threshold estimate and parameter estimates of equation (1).²¹ A comparison of Tables 2 and 6 reveals some interesting points. First, the threshold estimates are somewhat different but very close. The threshold estimates with yearly data are slightly higher for both industrial and developing countries (3 percent versus 1 percent for industrial countries, and 12 percent versus 11 percent for developing countries). Second, the high-inflation effect (that is, γ₂) is more powerful for yearly data. This is illustrated in the last three columns of Table 3. As expected, the fit is poorer with yearly data, but the threshold levels of inflation are precisely estimated.

Table 6.

NLLS With Fixed Effects

(Yearly Data)

Note: The panel has potentially 39 observations (T), covering 1960-98, for 140 countries (N). The variables are inflation, π; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the growth rate of terms of trade, dlog(tot). The dummy variable d^{π *} takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated country-specific effects are not reported.

Table 6.

NLLS With Fixed Effects

(Yearly Data)

Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{\pi *})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00054	0.00143	0.00043
	(-1.99)b	(3.22)a	(1.59)
$d^{\pi *}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.01180	-0.00923	-0.01347
	(-6.00)a	(-5.17)a	(-5.56)3
igdp	0.07820	0.02690	0.078M)
	(4.11)a	(0.92)	(3.83)a
dlog(pop)	-0.01557	-0.02750	-0.01701
	(-0.16)	(-.20)	1-0.17)
dlog(tot)	-4.72E-05	-0.01583	-1.77E-05
	(-0.00)	(-1.03)	(-0.00)
Threshold estimate (%)	9	3	12
	(8l.37)a	(25.24)a	(87.68)a
NxT	4264	950	3414
R2	0.14	0.50	0.12

Note: The panel has potentially 39 observations (T), covering 1960-98, for 140 countries (N). The variables are inflation, π; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the growth rate of terms of trade, dlog(tot). The dummy variable d^{π *} takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated country-specific effects are not reported.

View Table

Table 6.

NLLS With Fixed Effects

(Yearly Data)

Dependent Variable: dlog(gdp)
Independent Variables	All	Industrial	Developing
$(1-d^{\pi *})[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	0.00054	0.00143	0.00043
	(-1.99)b	(3.22)a	(1.59)
$d^{\pi *}[\log \left(\pi \right)-\log \left({\pi }^{*}\right)]$	-0.01180	-0.00923	-0.01347
	(-6.00)a	(-5.17)a	(-5.56)3
igdp	0.07820	0.02690	0.078M)
	(4.11)a	(0.92)	(3.83)a
dlog(pop)	-0.01557	-0.02750	-0.01701
	(-0.16)	(-.20)	1-0.17)
dlog(tot)	-4.72E-05	-0.01583	-1.77E-05
	(-0.00)	(-1.03)	(-0.00)
Threshold estimate (%)	9	3	12
	(8l.37)a	(25.24)a	(87.68)a
NxT	4264	950	3414
R2	0.14	0.50	0.12

Note: The panel has potentially 39 observations (T), covering 1960-98, for 140 countries (N). The variables are inflation, π; gross domestic investment over GDP, igdp; the growth rate of population, dlog(pop); and the growth rate of terms of trade, dlog(tot). The dummy variable d^{π *} takes one for inflation rates greater than the threshold estimate (π^*) and zero otherwise. The t-statistics, given in parentheses, are computed from White heteroskedasticity-consistent standard errors. The letters “a”, “b”, “c”, indicate statistical significance at 1, 5, and 10 percent, respectively. The growth rate of a variable x is approximated by the first difference of the log of x, dlog(x). The estimated country-specific effects are not reported.

The 95 percent confidence intervals for the whole sample, industrial countries, and developing countries, are [8.78, 9.22], [2.76, 3.24], and [11.80, 12.20], respectively. Considering the few number of observations with very low inflation rates for the five-year-averaged data, the 3 percent threshold estimate (versus 1 percent with smoothed data) for industrial countries may well be more reliable.

Sensitivity to Additional Explanatory Variables

As explained in Section I, only variables that were found to be robust in the empirical growth literature were included in the regression equation linking inflation to growth. The use of fixed effects also helps capture cross-country differences in GDP growth. Since endogenous growth theory has emphasized the role of human capital in the growth process of a country, equation (1) has been augmented by including a human capital variable. Following the empirical growth literature, human capital is proxied by enrollment rates in the primary, secondary, and tertiary schools.²² All three variables came out statistically insignificant. Furthermore, their inclusion does not significantly change the results. In fact, the threshold values remain the same. The reason may be that the three proxies (primary, secondary, and tertiary enrollment) are highly correlated with the initial income variable (ly₀). A regression of the former on the latter yields an R² of 0.98, 0.92, and 0.98, respectively. In other words, the initial income variable appears to be picking up most of the cross-country variation in school enrollment.

Financial development is another important variable that was emphasized by King and Levine (1993). Following this approach, we used three different proxies for financial depth. The first measures the size of the formal financial intermediary sector relative to economic activity (the ratio of liquid liabilities of the financial system, measured by M3 when it is available and M2 otherwise, to GDP); the second proxy measures the proportion of credit allocated to the private sector (the ratio of claims on the nonfinancial private sector to total domestic credit);²³ and the third is simply the second normalized by GDP instead of total domestic credit. Adding these variables did not change the estimated threshold values at all.