*5.2. Data Description*

The study utilizes panel data collected from the financial statements and annual reports of 227 non-financial enterprises listed at Ho Chi Minh Stock Exchange and Ha Noi Stock Exchange in Vietnam for the period 2008–2018. All these enterprises belong to di fferent manufacturing industries and thus, to capture their varying e ffects on the outcome, we perform the mix-e ffects regression. Time frequency indicates the year. The dataset has 1,974 observations. In Bayesian statistics, due to combining prior information with observed data, inferential results are valid to sparse data, and thus a small sample does not a ffect MCMC simulation results. It is noted that the 2008–2018 sample period includes years

2008–2009, when many countries around the world faced a sharp economic decline, but the Vietnamese enterprises were much less impacted by this global crisis. Statistical figures show that the economic growth of Vietnam achieved good performance, 5.7%, in 2008, and 5.4% in 2009 (World Bank 2019). Net revenue and fixed assets represent the enterprises' output and capital variables. The figures of net revenue and fixed assets are calculated based on the 2010 production price index of the General Statistics O ffice. The units of net revenue, fixed assets and labor are million VND, million VND and number of employees, respectively. The nonfinancial enterprises are chosen for our analysis since this sector is a powerful engine of Vietnamese economic growth, so to a grea<sup>t</sup> extent it stands for the national production. Moreover, according to Karabarbounis and Neiman (2014), the use of data on the enterprises listed on the stock market allows labor and capital shares not to be skewed owing to statistical errors that often occur when we take into account the mixed incomes from households' labor and capital contributions as well as those in the state-owned sector which are di fficult to be measured accurately. The measurements of the variables are presented in Table 1.



## **6. Empirical Results**

## *6.1. Descriptive Statistics*

Table 2 shows that variables *y2010*, *l*, and *k2010* obtain maximum value of 4.00 × 107, 19,828 and 2.27 × 107, minimum value of 5320, 17 and 270, mean of 1,519,804, 1186 and 497,570, respectively. Standard deviation (Std. Dev) measures the variation or dispersion of a set of values. It equals 3,516,699, 1793 and 1,614,555 for *y2010*, *l* and *k2010*, respectively.


#### *6.2. Bayesian Simulation Results*

Acceptance rate and e fficiency are two criteria for evaluating the e fficiency of MCMC sampling in Bayesian models. The acceptance rate is the number of proposals accepted in the total number of proposals, while e fficiency means the mixing properties of MCMC sampling. Both of these rates influence MCMC convergence. The simulation results demonstrate that our model has a high acceptance rate of 0.6. According to Roberts and Rosenthal (2001), acceptance rates between 0.15–0.5 are optimal. Therefore, the MCMC sampling of our regression model has reached an acceptable acceptance rate. The smallest, average and largest e fficiency of the MCMC sampling is 0.044, 0.21 and 0.97, which are greater than the warning level of 0.01 (Table 3). The MC errors (MCSE) of the posterior mean estimates are close to one decimal. The smaller these values are, the more accurate the estimates. In Bayesian analysis, posterior confidence intervals, as stated above, have a straightforward probability interpretation. For example, for our model, the probability of the posterior mean of the parameter β0 in the range (10.7; 11.2) is 95% (Table 3).



Random intercepts for *<sup>u</sup>*1*j* (id) denote the varying effects of 227 enterprises studied on the outcome of the model. Means of all the random effects ge<sup>t</sup> MCSE close to one decimal, which is reasonable for MCMC algorithms. For illustration, we demonstrate the random intercepts of the first 10 enterprises in Table 4.

**Table 4.** Estimated random effects of the first 10 enterprises.


#### *6.3. Convergence Test for MCMC Chains*

The convergence of MCMC chains should be tested before Bayes inference is performed, because Bayesian inference is robust only when the MCMC chains converge to a stationary distribution. According to the results recorded in Figure 1, with respect to our model, the diagnostic graphs are reasonable. Trace plots exhibiting no trends, run relatively quickly through the distribution towards the constant values of mean and variance; the autocorrelation plots are acceptable; histograms resemble the shape of probability distributions (Figure 1). In general, MCMC chains of our model have good mixing. Therefore, it can be concluded that there is no serious convergence problem and the MCMC chains have converged to the target distribution.

**Figure 1.** Graphical convergence diagnostics.

In addition, cusum plots are also a visual method for inspecting MCMC convergence. In our case, the cusum lines are not smooth but jagged, which surely points to MCMC convergence (Figure 1).

Besides visual inspection, formal test in which effective sample size can be used is a common method (Table 5). Efficiency greater than one is suggested satisfactory. Results presented in Table 5 demonstrate no sign of a non-convergence problem since the efficiency of all the model parameters is more than 4, whereas the highest correlation time is 22 lags.



#### *6.4. Estimation Result of the ES*

According to the results shown in Table 3, our estimated CES function has the value of efficiency parameter β0 = 10.9, a distribution parameter of δ = 0.7, and a substitution parameter of θ = 1.9. The Bayesian simulations do not provide point estimates in a frequentist sense. Tests for MCMC

convergence allow to confirm whether or not estimation results are robust. In our work, we already performed the convergence diagnostics, which produced acceptable results, as shown in the above. Once Bayesian inference is valid, MCMC iterations do yield similar estimates of the model parameters. These estimates point to the properties of a neoclassical production function. Because θ > 0, the ES is smaller than one (0 <σ< <sup>1</sup>). These empirical results coincide with most of previous studies (for example, Berndt 1976; Hamermesh 1993; Pereira 2003; Chirinko 2008; Young 2013). In case σ < 1, we can provide two main explanations for the Vietnamese nonfinancial enterprises' output growth.

First, our data set used in this study indicates that there is a marked di fference between the growth rates of capital and labor. Hence, with the ES lower than one, the sign of (12) is negative. Based on this finding, it can be concluded that the output growth rate of the Vietnamese nonfinancial enterprises has a falling trend in the long run. We should note that compared to enterprises in advanced economies, the Vietnamese ones have a very low contribution of technical change to production, and hence they are not capable of generating the unbounded endogenous growth. Therefore, stimulating R&D activities in enterprises is extremely important.

Second, as *gl* < *gK* and *K L* > 1, the higher growth rate of output is associated with a larger ES, i.e., <sup>δ</sup>*gy* δσ > 0. According to our result, the ES is less than one, so capital as a relatively scarce factor strongly influences the output since its elasticity of production is grea<sup>t</sup> (≈0.73). While the ES is rising, the elasticity of production will be diminishing for the capital, but it will increase for the labor. Under the current conditions of the Vietnamese economy, capital is a scarce factor of the economy, so substantially increasing investment should be one of the most significant growth policies. Specifically, it is necessary to attract more foreign direct investment and expand positive spillover e ffects from foreign corporations to the national enterprises.
