Going clubbing in the eighties: convergence in manufacturing sectors at a glance

Abstract

This paper employs the distribution dynamics framework for assessing labour productivity convergence, in the period 1980–1995, among 28 developed and developing countries, in different manufacturing compartments, identified as according to their research and development intensity. Three competing hypotheses are considered: absolute, conditional and club convergence. The key result of the analysis is twofold. First, consistently with very recent evidence, absolute convergence is found in manufacturing as a whole. Second, convergence tendencies are sector specific. In particular, club convergence characterizes traditional and medium- technology compartments, while the absolute one qualifies high-tech productions. Overall, these findings support the view that cross-country labour productivity convergence might be hindered by the sub-optimal structural reallocation from nonconvergence to convergence activities. Moreover, as the clustering dynamics in traditional and medium-tech sectors is related either to physical capital stock or technological development, laggard economies should purse ad hoc catching-up strategies. Finally, the result of high tech provides supportive evidence for the theory of dynamic comparative advantages. Thus, it seems desirable that emerging countries enter into technology-intense markets and that they develop the necessary capabilities for exploiting such endogenous advantages.

Appendices

Data appendix

1.1 Labour productivity

This is the natural logarithm of relative labour productivity. Labour productivity is measured as manufacturing value added per worker in 1996 PPP dollars. Relative labour productivity consists in the natural logarithm of labour productivity in country i, sector j, at time t relative to the one of the USA (i.e. the leader) in the same sector and period, which is formally written as $y_{ijt}=\hbox {log}(Y_{ijt}/Y_{USjt})$. Normalizing the data is important for removing some of the trend from the cross section and thus for avoiding degenerate long-run distributions. The choice of the normalizing variable was made following Quah (1996a) and Desmet and Fafchamps (2006), although other indicators could have been used (e.g. cross-country average, range, etc).

Sectoral relative labour productivity data were retrieved combining UNIDO Industrial Statistics Database 2004, at 3-digits of ISIC Code (Revision 2) (i.e. INDSTAT3), World Bank, World Development Indicators 2006 (i.e. WDI) and the Penn World Tables 2002 (i.e. PWT 6.1).²⁶

1.2 Steady state proxies

1.2.1 Log-relative sectoral investment rates in physical capital

They refer to the natural logarithm of gross fixed capital formation (i.e. GFCF) share to manufacturing value added in country i, sector j, at time t relative to the one of the USA. Relevant series come from the aforementioned UNIDO dataset.

1.2.2 Log-relative investment rates in human capital

They consist in the natural logarithm of the average years of schooling in the population over age 15 in country i at time t relative to the one of the USA. Data come from Barro and Lee (2013), which is the most up-to-date version of the well- acknowledged dataset. In particular, as original series are recorded at 5-year intervals, they were interpolated assuming a linear pattern. Moreover, population over 15 years was preferred to the one over 25 years because working age in developing countries can be quite low, Bennell (1996).

1.2.3 Development dummy

This is a dichotomous variable having as reference group the high-income economies, as defined in WDI.

1.3 Relevant initial conditions

1.3.1 Log-relative physical capital stock per worker

This is the natural logarithm of physical capital stock in country i, sector j, at time t relative to the one of the USA in the same sector and period, which is formally written as $k_{ijt}=\hbox {log}(K_{ijt}/K_{USjt})$.

Since sectoral physical capital stock series are not available from international data sources, they were estimated applying to the UNIDO GFCF series the perpetual inventory method assuming an exponential depreciation rate of 6 %. This is a quite standard hypothesis, see, for example, Vandenbussche et al. (2006) and Jones (1997).

As customary in the literature (Vandenbussche et al. 2006; Caselli 2005 and Young 1995), initial capital stock was calculated employing the formula of steady state capital stock in the neoclassical growth model. Thus, initial capital stock is written as $K_{t-1}=\frac{I_t}{g+0.06}$.

As UNIDO GFCF data start in 1960, $g$ is the output growth rate in the first 5 years available (i.e. 1960–1965), $K_{t-1}$ is the estimated capital stock in 1959 and $I_t$ is the actual investment in 1960. Since the present econometric exercise begins in 1980, the measurement error on this initial value should have disappeared (Vandenbussche et al. 2006; Nehru and Dhareshwar 1993).

The estimated sectoral capital stock is denominated in 1996 PPP Dollars.

To conclude, it is important to make a note of caution on the use of the same depreciation rate of capital for different sectors and for different countries. That different levels of technological progress should imply also different levels of depreciation rates is well documented in the literature, see Arslanalp et al. (2010). Thus, as robustness checks, capital stock series were calculated employing a 4 % and a 15 % depreciation rate, respectively, used by Nehru and Dhareshwar (1993) and Caselli and Wilson (2004). The findings of the distribution dynamics exercise are qualitatively unaltered. The robustness checks are available upon request.

1.3.2 Interacted TFPgap

Following Griffith et al. (2004), sector- and year-specific TFPgap is calculated as the difference between leader’s TFP and the one of any other country, where TFP levels are obtained through the superlative index approach of Caves (1982) and Caves et al. (1982), see the Appendix on TFP estimation for full details. The natural logarithm of secondary schooling attainment rate in any country i normalized with respect to the one of United States serves, instead, as absorption capacity proxy.

Appendix on TFP estimation

Total factor productivity (TFP) or Solow residual is the part of output growth not accounted for market transactions. It originates from growth accounting exercise, and it is conventionally employed to measure technological progress. Following Diewert (1976), Caves et al. (1982) derive an index number that allows TFP comparisons among countries. This index is superlative, meaning that is exact for the flexible aggregator function chosen (i.e. translog production function), and transitive, so that the choice of base country and year is inconsequential.²⁷

Formally, it is assumed that value added of a generic country i is a function of capital stock and employment, that is translog with identical second-order term, that constant returns to scale apply and that inputs are measured perfectly and in the same units for each observation. In symbols:

$$\begin{aligned} \ln y_{i}=\alpha _{0i}+\alpha _{1i}lnl_{i}+\alpha _{2i}lnk_{i}+\alpha _{3}(lnl_{i})^{2}+ \alpha _{4}(lnk_{i})^{2}+\alpha _{5}(lnl_{i}*lnk_{i}) \end{aligned}$$

where constant returns to scale hypothesis requires $\alpha _{1i}+\alpha _{2i}=1$ and $2\alpha _{3}+\alpha _{5}=2\alpha _{4}+\alpha _{5}=0$.

In this Appendix, Caves et al. (1982) contribution is reviewed beginning with TFP index number for bilateral comparisons.

There are two countries, b and c, country b is the basis of comparison and the distance function $D_{c}(y_{b},l_{b},k_{b})$ represents the minimum proportional decrease in $y_{b}$ such that the resulting output is producible with the inputs and productivity levels of c. Or $D_{c}(y_{b},l_{b},k_{b})$ is the smallest input bundle capable of producing $y_{b}$ using the technology in country c. In symbols:

$$\begin{aligned} D_c(y_{b},x_{b})=\min \left\{ \delta \in \mathfrak {R}_{+} : f_c (\delta x_{b})\ge y_{b} \right\} \end{aligned}$$

where $x_{b}=\left( k_{b},l_{b}\right)$.²⁸ Assuming that producers are cost minimizers and price takers in input markets, it can be shown that the Malmquist index (i.e. the geometric mean) of two distance functions for any two countries c and b gives the following TFP index:

$$\begin{aligned} \hbox {TFP}_{cb}=\frac{y_{c}}{y_{b}}\left( \frac{\bar{l}}{l_{c}}\right) ^{\sigma _{c}} \left( \frac{\bar{k}}{k_{c}}\right) ^{1-\sigma _{c}}\left( \frac{l_{b}}{\bar{l}} \right) ^{\sigma _{b}}\left( \frac{k_{b}}{\bar{k}}\right) ^{1-\sigma _{b}} \end{aligned}$$

where a bar denotes an average over countries and $\sigma _i=\left( \alpha _{i}+\overline{\alpha }\right) /2$, where $(\alpha _{i})$ stands for labour’s share in total costs for country $i$.

Similar reasoning can be applied to derive the multilateral version of TFP index, which allows for TFP comparisons among more than two countries. Taking sectoral heterogeneity explicitly into account, TFP level in country $i$, sector $j$ at time $t$ is

$$\begin{aligned} \hbox {TFP}_{ijt}=\frac{Y_{ijt}}{\overline{Y_{jt}}}\left( \frac{\overline{L_{jt}}}{L_{ijt}} \right) ^{\tilde{\sigma }_{ijt}}\left( \frac{\overline{K_{jt}}}{K_{ijt}} \right) ^{{1-\tilde{\sigma }}_{ijt}} \end{aligned}$$

where a bar denotes the geometric average over all countries for a given sector $j$ and a year t and ${\tilde{\sigma }_{i j t}}=(\alpha _{ijt}+\overline{\alpha _{j}})/2$, where ${\alpha _{ijt}}$ is labour share in country i and industry j and $\overline{\alpha _{j}}$ is the cross-country average for industry j.

Then, taking natural logarithms, the previous expression becomes:

$$\begin{aligned} \hbox {TFP}_{ijt}=\ln \left( \frac{Y_{ijt}}{\overline{Y_{jt}}}\right) -\tilde{\sigma }_{ijt} \ln \left( \frac{L_{ijt}}{\overline{L_{it}}}\right) -(1-\tilde{\sigma }_{ijt}) \ln \left( \frac{K_{ijt}}{\overline{K_{jt}}}\right) \end{aligned}$$

As originally noticed by Harrigan (1997), the variability in actual labour shares over value added makes difficult the empirical implementation of the equation above. To solve this problem, smoothed and not actual labour shares are usually employed.

Smoothed labour shares are simply obtained running a regression of actual labour shares on a constant and the capital to labour ratio:²⁹

$$\begin{aligned} \alpha _{ijt}=\xi _{i}+\xi _{j}+\chi _{ij}\ln \left( K_{ijt}/L_{ijt}\right) \end{aligned}$$

Previous studies on developed countries, such as Harrigan (1997) and Griffith et al. (2004), consider only sectoral heterogeneity in slopes (i.e. $\chi _{j}$). As the sample of countries employed in the current contribution comprises developing countries, the original specification has been improved allowing for country and sector heterogeneity in both intercepts and slopes (i.e. $\xi _{i}$, $\xi _{j}$ and $\chi _{ij}$). In particular, to avoid a major loss in data variability, due to many dummies, manufacturing sectors have been grouped as according to Lall’s taxonomy and the sampled economies have been divided into developed and developing ones, using World Bank definitions. The diagnostics employed strongly reject the null hypothesis of nonheterogeneity in both intercepts and slopes among different sectors and countries. More precisely, using panel data F tests, intercept heterogeneity due to country and sector-fixed effects has been detected. Sector and country heterogeneity, in both slope and intercepts, has been confirmed through Chow type F statistics.³⁰

Appendix on distribution dynamics

1.1 Distribution dynamics and conditioning: a brief nontechnical summary

When distribution dynamics is employed, convergence tendencies among countries can be retrieved analysing the evolution along time of cross-country labour productivity distribution. In particular, the main question to be answered is whether all economies considered converge to same level of labour productivity, such that the cross-country distribution is single-peaked, or whether the economies converge only within small clubs, such that the distribution exhibits more than one peak.

Operatively, the changes along time of cross-country labour productivity distribution are retrieved using the stochastic kernel density estimator. In fact, this estimator allows to measure the probabilities of dynamic transitions from one labour productivity class to another, for each economy.

Intuitively, the stochastic kernel can be thought as a refinement of the histogram. In particular, while in histogram the frequency distribution is calculated for disjoint states, with kernel density estimator, the frequency distribution is estimated for a large number of overlapping class intervals, which gives a much smoother appearance, resembling a probability density function.

Two are the types of kernels employed in this paper:

• 1. unconditioned kernels

• 2. conditioned kernels

The unconditioned kernels give information on the likelihood that an economy, starting from a given relative position in the initial period $t$, will end up improving or worsening its relative position in the final period $t+s$. In other words, it can be said that unconditioned kernels measure the transition probabilities from $t$ to $t+s$.

Unconditioned kernels are used here to test the absolute convergence hypothesis.

Conditioned kernels are an extension of unconditioned ones. In particular, they allow to identify the factors that eventually lead club convergence dynamics. The effects of conditioning are identified by changes in shape and location of the kernel, with respect to the unconditioned case.

Conditioned kernels are here employed for testing both conditional convergence hypothesis and club convergence determinants.

In the case of conditional convergence, for example, if the unconditioned kernel shows twin-peaks feature and, after conditioning with respect to steady state proxies, it is found that the conditioned kernel is single-peaked, then it can be said that club dynamics is lead by structural differences and that conditional converge hypothesis is not rejected.

1.2 Unconditioned transition probability estimates

This section provides a technical illustration of the methodology employed to estimate unconditioned transition probabilities, which are used to test the absolute convergence hypothesis.

Sectoral convergence tendencies are inferred analysing the dynamic behaviour of cross-country distribution of log-relative labour productivity.³¹

Individual country $i$ labour productivity, in sector $j$, at time $t$ is called $y_{it}$, where the sector index has been omitted for notational convenience (i.e. $y_{it}=log(Y_{ijt}/Y_{USjt})$). Cross- country, sector-specific, labour productivity distribution, at time $t$, is denoted as $f_{Y_t}(y_t)$, where $Y_t$ indicates the corresponding random variable.

It is assumed that year-to-year changes in the distribution of labour productivity can be represented by an homogeneous Markow process, in such a way that, $\forall t$:

• 1. $f_{Y_{t+1}|Y_t}(y_{t+1}|y_t)=f_{Y_{t+1}|Y_t}(y_{t+1}| y_t,y_{t-1},y_{t-2},...)$

• 2. $f_{Y_{t+1}|Y_t}(y_{t+1}|y_t)=f_{Y_t|Y_{t-1}}(y_{t}|y_{t-1})$

The first property guarantees that only previous period income distribution impacts on next period one (i.e. history does not matter). The homogeneity assumption in 2 ensures that the transition probabilities do not vary with time. Although quite restrictive, both hypotheses are necessary for estimating long-run transition probabilities given the available data.

Conditional density functions, $f_{Y_{t+1}|Y_t}(y_{t+1}|y_{t})$, represent the cornerstone of distribution dynamics convergence analysis. This kind of distribution, in fact, encodes information about individual economies’ passages over time. Thus, it sheds light on both intra-distribution dynamics and external shapes, making inference about convergence tendencies possible. For example, observing conditional density mappings, is it possible to know whether poor countries are catching up with their richer counterparts, whether rich countries are still enriching, whether countries are converging overall or are clustering within clubs.

The empirical estimation of conditional densities is handled by nonparametric techniques. To begin, it is worth to recall the definition of conditional distribution, that is the joint distribution divided by the marginal distribution. In formal terms:

$$\begin{aligned} f_{Y_{t+1}|Y_t}(y_{t+1}|y_{t})= \frac{f_{Y_{t+1},Y_t}(y_{t+1}, y_{t})}{f_{Y_t}(y_{t})} \end{aligned}$$

(1)

The joint distribution of $(Y_{t+1}, Y_{t})$ can be estimated nonparametrically using a bivariate stochastic kernel, while the marginal distribution of $Y_{t}$ is obtained by numerical integration of the joint distribution. Finally, the conditional distribution is simply obtained by dividing one to the other, after appropriate discretization of the joint support.³²

Long-run tendencies towards convergence are encoded by the ergodic distribution. This is the stationary distribution of labour productivity, which will be approached in the long run should certain technical conditions hold.³³ In particular, if the ergodic distribution is unimodal and has a low variance, then long-run cross-country convergence can be claimed.

Formally, the ergodic is the distribution $f$ which solves the following functional equation:

$$\begin{aligned} f(y_{t+1})=\int _{-\infty }^{+\infty }f_{Y_{t+1}|Y_t}(y_{t+1}|y_{t})f(y_{t})dy \end{aligned}$$

(2)

In order to compute the ergodic distribution, the support of $y$ is discretized in a set of $N$ equally large intervals, where interval $h$ is denoted as $\varOmega _{h}$.³⁴

Then, the probabilities of transition from one interval to another are calculated. Formally, the probability of transition from the interval $\varOmega _{h}$ to another, $\varOmega _{k}$, in one time period, is denoted as

$$\begin{aligned} \alpha _{hk}=Pr(y_{t+1}\in \varOmega _{k}|y_{t}\in \varOmega _{h}) \end{aligned}$$

At this point of the explanation, it is useful to adopt a compact matrix notation. Hence, the ergodic distribution is the vector $p$ that solves the following system of equations:

$$\begin{aligned}&\displaystyle p=Ap\\&\displaystyle (I-A)p=0 \end{aligned}$$

where each component of the vector $p$ represents the probability of $y$ assuming a value comprised in a given $\varOmega$ and A is the matrix of transition probabilities $\alpha _{hk}$.

Since each column of matrix A is a marginal density and then its elements sum to 1, A does not have full rank, and by consequence, the system does not have a unique solution. To find a unique solution, it is standard to simply drop one row of A (to make its columns linearly independent) and then add the restriction that the entries of vector $p$ sum to 1.³⁵ Then, matrix A is rewritten as B

The modified system is then

$$\begin{aligned} Bp=b \end{aligned}$$

where the vector $b$, for the constraint added, has all entries equal to 0 except the last one, which is equal to 1.

At this point, the unique ergodic distribution, $p$, can be easily found inverting B:

$$\begin{aligned} p=B^{-1}b \end{aligned}$$

1.3 Conditioning techniques

This part outlines the conditioning technique employed to test for conditional convergence and club convergence determinants.

Under the conditional convergence hypothesis, cross-country productivity equalization cannot be found in the original relative labour productivity distribution, $f_{Y}$, but in the conditioned one, $f_{Y|X}$, where $X$ denotes steady state proxies. Then, the object of interest is the transition probabilities of the part of labour productivity not explained by the auxiliary variables (i.e. steady state proxies). Employing the former notation, such transition probabilities are formally written as

$$\begin{aligned} f_{Y_{t+1}|Y_t,X_t}(y_{t+1}|y_{t},x_t) \end{aligned}$$

(3)

Exploiting Chamberlain (1984) results, the part of labour productivity orthogonal to auxiliary variables is computed as ordinary least squares (OLS) residuals of the projection of labour productivity growth on each of the steady state proxies.³⁶ Such calculation involves three steps:

• 1. estimating the part of countries’ relative productivity growth rate explained by conditioning steady state variables;

• 2. finding the initial level of relative labour productivity explained by conditioning steady state variables;

• 3. combining the previous results to find the level of relative labour productivity unexplained by the auxiliary variables (i.e. orthogonal to steady state proxies).

Call $g_{it}$ the growth rate of $y_{it}$ (i.e. log-relative productivity in country $i$, sector $j$ at time $t$), where again the sector index is omitted for notational convenience. Name $\widehat{g_{it}}$ the part of $g_{it}$ explained steady state proxies, which are investment rate in both physical and human capital, indicated as $r_{it}$ and $h_{it}$, and the dummy development, $ddev$. Finally, the part of labour productivity orthogonal to steady state proxies, which is the object of interest, is called $\widehat{\epsilon }_{it}$.

Step 1 is implemented regressing $g_{it}$ on a two-sided distributed lag of conditioning variables and saving the fitted values. One of such regressions is run for each steady state proxy. Then, cumulating the fitted values, by country and sector, the part of countries’ relative productivity growth rate explained by conditioning steady state variables, $\widehat{g_{it}}$, is obtained.

Note that in empirical work, multisided regressions are employed to handle endogeneity issues, which are represented in this specific case by the likely bidirectional causality between labour productivity growth rate and steady state proxies. This technique, introduced by Sims (1972), has been extensively used by Quah, who noticed that just 2 leads and 2 lags are sufficient to clear the estimated growth rate from feedback effects, Quah (1996a).

Step 2 is taken running a pooled OLS regression of $y_{it}$ on steady state proxies’ time averages (i.e. $\overline{r_{it}}$ and $\overline{h_{it}}$) and the estimated growth rate (i.e.$\widehat{g_{it}}$). For each sector, the coefficients that solves the following minimization problem are used to pin down the initial level of labour productivity explained by steady state variables, $\widehat{y_{i0}}$:³⁷

$$\begin{aligned} min_{\beta _{1},\beta _{2},\beta _{3}}\sum _{i}\sum _{t}[y_{it}-(\beta _{1} \overline{r_{it}}+\beta _{2}\overline{h_{it}}+\beta _{3}ddev+ \widehat{g_{it}})]^{2} \end{aligned}$$

In fact, thanks to the estimated coefficients, $\widehat{\beta }s$, the initial level of log-relative labour productivity explained by conditioning variables can be expressed as

$$\begin{aligned} \widehat{y_{i0}}=\widehat{\beta _{1}}\overline{r_{it}}+\widehat{\beta _{2}} \overline{h_{it}}+\widehat{\beta _{3}}ddev \end{aligned}$$

Then, adding the growth rates of step 1, the level of relative labour productivity explained by steady state variable is calculated as

$$\begin{aligned} \widehat{y_{it}}=\widehat{y_{i0}}+\widehat{g_{it}} \end{aligned}$$

Finally, $\epsilon _{it}$, which represents the productivity level not accounted for (or conditional to) steady state proxies, is simply found subtracting the estimated relative labour productivity from the actual one:

$$\begin{aligned} \widehat{\epsilon }_{it}=y_{it}-\widehat{y_{it}} \end{aligned}$$

Once country- and sector-specific $\epsilon _{it}$ series have been calculated, the empirical implementation for testing conditional convergence is the same as absolute (or unconditional) convergence.

In particular, bivariate stochastic kernel densities fit the cross-country, sector-specific distribution of relative productivity orthogonal to steady state variables, which is denoted as $f_{\widehat{E_{t+1}},\widehat{E_{t}}}(\widehat{\epsilon _{t+1}},\widehat{\epsilon _{t}})$. By numerical integration of the joint distribution, the marginal density $f_{\widehat{E_{t}}}(\widehat{\epsilon _{t}})$ is obtained. Finally, the transition probabilities of Equation (3) are found dividing the joint distribution, $f_{\widehat{E_{t+1}},\widehat{E_{t}}}(\widehat{\epsilon _{t+1}},\widehat{\epsilon _{t}})$, by the marginal distribution, $f_{\widehat{E_{t}}}(\widehat{\epsilon _{t}})$.

Long-run distribution of relative labour productivity conditioned to steady state variables is retrieved from the ergodic distribution of random variable $\widehat{\epsilon _{t}}$. Such a distribution is calculated as for the unconditional case (previous section).

Turning now to club convergence analysis, it should be intuitive that the conditioning scheme described so far can be easily extended to determine the relative strength of club convergence inner drivers.

In particular, when club convergence hypothesis holds, the object of interest becomes the dynamics of labour productivity distribution conditioned to both steady state proxies and club convergence driving forces, namely capital and technological initial conditions. Formally, the following transition probabilities has to be computed:

$$\begin{aligned} f_{Y_{t+1}|Y_t,X_t,Z_t}(y_{t+1}|y_{t},x_t,z_t) \end{aligned}$$

(4)

where the variable $Z$ represents either initial capital stock or initial technological level.

To retrieve the relative strength of capital stock (or technology) as club convergence determinant, relative labour productivity orthogonal to both steady state proxies and capital stock (or technology) initial level must be calculated. This is done implementing the three steps previously described, taking into consideration capital stock (or technology) as extra conditioning variable.

By the same tokens as before, the density in Equation (4) and the ergodic distributions are computed.

To conclude, it is worth noticing that the conditioning scheme here employed allows not only to work out alternative convergence hypotheses within a unified framework but also to calculate the ergodic of distributions that have been conditioned to time varying (and likely endogenous) variables. This latter aspect is particularly worth because it represents a step forward with respect to long-run convergence analysis based on both discrete transition probability matrices, such as in Quah (1996a, 1997), Epstein et al. (2007) and Maffezzoli (2006), and time-invariant conditioning factors such as in Desmet and Fafchamps (2006).

1.4 Statistical inference

The reliability of the kernel density estimations presented in this paper has been checked through a number of statistical inference routines.³⁸

First, following Salgado-Ugarte et al. (1995) and their survey on different methods for selecting the bandwidth for univariate density estimation, Silverman (1986) Gaussian kernel optimal bandwidth has been calculated for each assessed convergence hypothesis, for all manufacturing sectors under consideration.³⁹

Second, drawing from the work of Fiorio (2004), the asymptotic $95\%$ confidence intervals for kernel density estimation, based on the theory of kernel density confidence intervals estimation developed in Hall (1992), have been calculated employing Silverman’s Gaussian kernel optimal bandwidth.⁴⁰

Third, the number of modes of the estimated kernel densities as well as their values has been assessed following the ASH-WARPing procedure, Scott (1992) and Haerdle (1991), employing as optimal bandwith the one of Silverman.⁴¹

Finally, the nonparametric assessment of multimodality has been done following the work of Silverman (1981) as described in Salgado-Ugarte et al. (1997). In particular, the test proposed by Silverman uses nonparametric kernel density estimation techniques to determine the most probable number of modes in the underlying density univariate data, employing bootstrapped samples. Silverman’s test reports the $p$ value associated with each number of modes for each bootstrapped sample. As such, a testing procedure should be repeated for a successively larger number of modes until a sufficiently large $p$ value is obtained, one common rule of thumb is stopping the test once a nominal $p$ value of 0.40 has been found, Salgado-Ugarte et al. (1997).⁴²