*3.3. Models and Estimation Methods*

This study aimed to test whether there is an optimal threshold between the cash holding ratio and company's performance. According to the trade-off theory, the optimal ratio of cash holdings is determined by a trade-off between marginal cost and profit margin of cash holdings (Opler et al. 1999). Therefore, this study assumed the existence of an optimal ratio of cash holdings, and tried to use the threshold regression model to estimate this ratio. To test the hypothesis, this study applied the threshold regression model of Hansen (1999). Single-threshold and multi-threshold models were based on the threshold regression model of Hansen (1999) as follows.

The single-threshold regression model was shown as:

$$\text{ROA}\_{\text{i,t}} = \begin{cases} \mu\_{\text{i}} + \theta' \text{H}\_{\text{i,t}} + \beta\_1 \text{CASH}\_{\text{i,t}} + \varepsilon\_{\text{i,t}} \text{ if } \text{CASH}\_{\text{i,t}} \le \gamma\\ \mu\_{\text{i}} + \theta' \text{H}\_{\text{i,t}} + \beta\_2 \text{CASH}\_{\text{i,t}} + \varepsilon\_{\text{i,t}} \text{ if } \text{CASH}\_{\text{i,t}} > \gamma \end{cases} \tag{6}$$

where θ- = (θ1, θ2, θ3) and Hi,t = (SIZEi,t, MBi,t, LEVi,t); ROAi,t represents for firm's performance, measured by profit before tax and interest on total assets; CASHi,t represents the proportion of cash held by the company, measured by the ratio of cash and cash equivalents on total assets; (CASHi,t) is the explanatory variable and also the threshold variable, estimated at each different threshold; Hi,t are control variables that affect company performance, including company size (SIZEi,t), company growth (MBi,t) and leverage (LEVi,t); θ1, θ2, and θ<sup>3</sup> are the estimated regression coefficients of the corresponding control variables; γ is the value of the estimated threshold; μ<sup>i</sup> is a fixed effect representing the heterogeneity of companies operating under different conditions; β<sup>1</sup> and β<sup>2</sup> are regression coefficients of the proportion of cash held by the company; the error εi,t is assumed to be independent and has a normal distribution <sup>ε</sup>i,t <sup>∼</sup> iid(0, <sup>σ</sup>2) ; i represents different companies; t represents different periods.

According to Hansen (1999), for estimating procedures, this study first removed the fixed effect (μi) by using the techniques of estimating "internal transformation" in a traditional fixed-effects model for panel data. By using the ordinary least square estimation method and minimizing the sum of squared error (S1(γ)), the test can obtain the estimation of the threshold value (γˆ) and the residual variance (σˆ <sup>2</sup>).

For testing procedures, this research first tested the hypothesis that there is no threshold effect (H0 : <sup>β</sup><sup>1</sup> = <sup>β</sup>2), using the likelihood ratio: F1 = (S0 <sup>−</sup> S1(γˆ))/ ˆσ2, where S0 and S1(γˆ) are the sum of squared error under hypothesis H0 and the opposite hypothesis (H1 : β<sup>1</sup> β2), respectively. However, since the asymptotic distribution of F1 is not normal, we used the bootstrap procedure to determine critical values and probability values (*p*-value). When the *p*-value is less than the desired condition value, we reject the H0 hypothesis.

When a threshold effect exists (β<sup>1</sup> β2), Hansen (1999) considered that γ is consistent with γ<sup>0</sup> (actual value of γ) and asymptotic distribution is not normal at a significant level. Therefore, we needed to check the asymptotic distribution of the estimated threshold with the hypothesis H0 : γ = γ0, by applying the likelihood ratio: LR1 = (S1(γ) <sup>−</sup> S1(γˆ))/ ˆσ<sup>2</sup> with asymptotic confidence intervals of <sup>c</sup>(α) = <sup>−</sup><sup>2</sup> log (<sup>1</sup> <sup>−</sup> <sup>√</sup> 1 − α), where α is the significance level (1%, 5%, and 10%). With the significance level α and LR1(γ0) > C(α), we can reject the hypothesis H0 : γ = γ0, meaning that the actual threshold value is not equal to the estimated threshold value.

If there exists a double threshold, the model can be modified as follows:

$$\text{ROA}\_{\text{i,t}} = \begin{cases} \mu\_{\text{i}} + \theta' \text{H}\_{\text{i,t}} + \beta\_1 \text{CASH}\_{\text{i,t}} + \varepsilon\_{\text{it}\prime} & \text{if } \text{CASH}\_{\text{i,t}} \le \gamma\_1\\ \mu\_{\text{i}} + \theta' \text{H}\_{\text{i,t}} + \beta\_2 \text{CASH}\_{\text{i,t}} + \varepsilon\_{\text{it}\prime} & \text{if } \gamma\_1 < \text{CASH}\_{\text{i,t}} \le \gamma\_2\\ \mu\_{\text{i}} + \theta' \text{H}\_{\text{i,t}} + \beta\_3 \text{CASH}\_{\text{i,t}} + \varepsilon\_{\text{it}\prime} & \text{if } \text{CASH}\_{\text{i,t}} > \gamma\_2 \end{cases} \tag{7}$$

where γ<sup>1</sup> < γ2. This can be extended to multi-threshold models (γ1, γ2, γ3, ... , γ*n*).

According to Li (2016), in econometrics, the endogeneity problem arises when the explanatory variables and the error term are correlated in a regression model, leading to biased and inconsistent parameter estimates. Particularly, this problem plagues almost every aspect of empirical corporate finance. To solve for the endogeneity problem, among all the remedies, the generalized method of moments (GMM) has the greatest correction effect on the coefficient, followed by instrumental variables, fixed-effects models, lagged dependent variables, and control variables.

Earlier literature on corporate cash holdings showed that there exist problems of endogeneity and omitted variable bias (Ozkan and Ozkan 2004). The endogeneity problem might arise in cash literature for several reasons. For example, firm-specific characteristics are not strictly exogenous, and have shocks affecting firm performance as well as influencing dependent variable CASH like size and leverage. Additionally, the presence of dependent variables may be correlated with past and current residual terms. To solve for the endogeneity problem that appears in the empirical analysis of cash holdings and firm value, Martínez-Sola et al. (2013), Azmat (2014) and Nguyen et al. (2016) applied the dynamic regression model—GMM estimation.

The threshold regression methods by Hansen (1999) were developed for non-dynamic panels with individual specific fixed effects. Least squares estimation of the threshold and regression slopes was proposed using fixed-effects transformations. This method has the disadvantage that the independent variables in the model are exogenous assumptions, which may in fact be endogenous. Therefore, this method explicitly excludes the presence of endogenous variables, and this has been an impediment to empirical application, including dynamic panel models.

#### **4. Results and Discussions**

#### *4.1. Descriptive Statistics for Variables in the Model*

Table 1 below presents descriptive statistic for the variables in the model. All of these variables were calculated based on the financial information collected from the balance sheet and income statement of 306 non-financial companies listed on the Vietnam stock exchange market in the period of 2008–2017.


**Table 1.** Descriptive statistic of variables.

Note: ROA represents company performance, measured by profit before tax and interest on total assets; CASH represents the percentage of cash held by the company, measured by the ratio of money and cash equivalents to total assets; MB represents company growth, measured by market value over book value of stocks; SIZE represents company size, measured by Ln (total assets); LEV represents company leverage, measured by total debt over total assets.

The statistical results described in Table 1 showed that the average ROA is 9.55%, indicating that in average with 1 VND (VND commonly refers to Vietnamese đồng, the currency of Vietnam) of capital used annually, firms can generate about 0.0955 VND profit before tax and interest. The average cash holding ratio (CASH) is 10.49%, indicating that cash and cash equivalents account for 10.49% of the company's total assets. The average firm's size is 26.6859, equivalent to 389 billion VND, the ratio of market value to book value is 1.1012 on average, and the average leverage (LEV) is 48.52%. Observation numbers, median values, standard deviations, and minimum and maximum values of variables are also presented in Table 1.
