Investigating the association of psychological impacts of delay on the perceived importance of real-time information: three-stage least square instrumental regression

Abstract

This study was conducted to evaluate the association between psychological impacts of delay on commuters and three indictors of the rail transport real-time information. The case study experienced sporadic long delay and simultaneously imprecise real-time information. Three dependent variables related to the passenger information system (PIS) were modeled simultaneously, including perceived waiting time, satisfaction, and the quality of rail transport. Given the likely correlation between the considered models error terms, the relationship between dependent and the independent variables was modeled with the three-stage least square (3SLS) model. We found, for instance, that there are significant association between the  impacts of the real-time information and commuters’ gender, purpose of travel, frequency of travel, and various psychological feelings that commuters experienced. However, interaction results highlighted that the associations between experienced negative emotions and dependent variables are not stable but vary based on other predictors. For instance, the association between purpose of travel and increased satisfaction due to accurate real-time information varies based on the frequency of using the transport by commuters. Also, the association between feeling frustrated and increased perceived time due to imprecise real-time information differs based on the amount of anger the commuters experience due to delay. In summary, we found that commuters who experience a higher level of negative feelings are expected to assign a higher importance to the accuracy of real-time information, but those effect are varied based on their purpose of travels or other demographic characteristics.

1 Introduction

Public transportation systems can contribute to a healthier environment by reducing air pollution [1]. Efforts have been made to persuade commuters to give up their cars for the sake of public transport by enhancing traveler’s satisfaction with mode of public transport, which could be achieved by an improvement in the quality of a certain service including its reliability [2].

Despite the efforts to improve the quality of public transport, the transport service is not without shortcomings. An important aspect of the public transportation is the precision of its arrival [3]. We define delay as the difference between actual arrival of a public transportation and expectation of a commuter, where the expectation of commuters might be gained by real–time information display [4].

Transport companies exert their efforts to avoid delay and improve punctuality, as delays could lead to commuter dissatisfaction with public transportation. However, because delays are sometimes unavoidable, it is important to know how delay is translated to the commuters in terms of different psychological feelings, and consequently how those feelings might impact the perceived quality of the transport. Understanding those factors could help the policy makers to take appropriate actions to improve the perceived quality of the service.

Provision of the Passenger Information System (PIS), with an objective of improving the reliability of the transport service, has been one of the main efforts employed by the policy makers. The PIS offers the passengers with information in real-time about the arrivals of public transportation.

However, in some instances, despite having unreliable public transportation, the PIS might also provide unreliable information regarding the arrival of the public transport. For those scenarios, the impact of inaccurate information of PIS on psychological feelings of commuters, and how those feelings impact the quality of that public transportation are worth investigation.

For our case study, at the terminal of the station, the PIS was provided for the commuters by means of liquid crystal display (LCD) display. Despite having long and regular delay, due to imprecise performance of PIS, there was an uncertainty about the arrival of the rail transport. For instance, the system did not provide reliable information about the arrival of the train and was not updated frequently. Thus, this study considers various psychophysical feelings that commuters might experience to highlight psychophysical feelings’ associations with various aspects of the PIS.

1.1 Real-time information

Extensive research has been conducted on various aspects of the PIS. For instance, the importance of PIS or real-time information for bus commuters was evaluated [5]. The results show that multimodal commuters are more likely to consult several apps, while non-regular passengers are more likely to consult general-purpose apps. The impact of real-time information on commuters was evaluated in another study [6]. The results highlighted that those commuters obtaining traffic information from traffic applications are more likely to switch their routes in regard to different characteristics.

1.2 The impacts of delay on commuters

As very limited studies have examined the impact of real-time information, especially the impact of imprecise PIS, this section presents studies investigated the impacts of delay on various aspects of commuters and studies that evaluated the impact of PIS on commuters’ feelings.

The impact of delay on car commuters was evaluated in a traffic jam case study [7]. In that study, a higher level of stress was linked to those commuters who were having a higher level of uncertainty about their arrival time. In addition, travel time reliability or delay was found to be an influential factor for level of commuting satisfaction [8].

The impact of public transport real-time information on customers was evaluated in another study [9]. The positive psychological factors of real-time information were highlighted in that study including factors such as reduced uncertainty, greater feeling of security, and higher customer satisfaction due to real-time information. In another study, structural equation modeling (SEM) was used to evaluate how passengers view the PIS while undergoing various psychological feelings [10]. The links were found between commuters’ emotions and PIS. For instance, it was found that there is an indirect relationship between motion sickness and PIS through other physiological factors.

1.3 Multiple regression equations

In this study, we used three equations for three indicators, or dependent variables of the real-time information, so it is worth discussing the necessity of applying simultaneous multiple regression models. There are reasons that could set forward for justifying the use of simultaneous evaluation of multiple equations. The main reason is a great likelihood that the unseen factors impacting the error terms in one equation, impact the error term in other related equations, which could result in correlations across the error terms of the equations. Ignoring that correlation might lead to biased or inefficient estimations of models’ parameters.

In addition, to come up with a valid modeling result, it is important to account for a complex relationship between variables. In case of endogeneity or correlation, not accounting for those effects could lead to biased results. The multiple equations could account for endogeneity by considering the response in one equation, and as a predictor in another equation.

Finally, this study was conducted to fulfill the following objectives:

• In the instrument, we asked respondents questions regarding different characteristics of the real-time information. To account for all those question in a unified framework, we implemented seemingly unrelated regression (SUR). This technique could provide a more realistic relations between responses and variables.

• We also considered interactions terms to account for the complex relationship between response and predictors. For instance, does the feelings of commuters due to delay varies based on their age or the purpose of their travels?

• Demographic characteristics, purpose of travel and frequency of transport use were included to understand the impacts of various commuters’ characteristics on assigning importance to the real-time information.

• Although previous studies have evaluated various aspects of real-time information, this study is one of the earliest studies that considered the malfunctioned rail transport with imprecise real-time information. This is also one of the earliest studies that considered and linked various feelings due to shortcomings of the rail transport and perception regarding the service.

2 Data

In this study, various sources were employed for the design of psychological aspects of our instrument. For instance, one of our main sources was based on the self-report measure of stress [11]. In that study, the implemented scale included different physiological, and psychological descriptors. Those factors include psychological attributes such as being angry, nervous, or stressed. The emotions are also based on the past study [12], where fifteen different emotions were highlighted including emotions such as anger, anxiety, sadness, and disgust.

The physical section of the instrument was based on Cohen-Hoberman inventory of physical symptoms (CHIPS) [13]. That is a list of 39 common physical symptoms highlighting relationships between negative life stress and various physical symptomatology. Those include factors such as backpain, diarrhea, and headache.

The questionnaires were collected in one of the main stations of Keretapi Tanah Melayu (KTM) in Malaysia. In the current study, to evaluate various feelings that the commuters might experience due to delay, they were questioned in the instrument for instance “I get angry when I face delay in KTM” and the responses vary from 1-strongly agree to 5-strongly disagree.

To tackle language barriers, questionnaires were translated into the local language, Malay. An introduction was included in the questionnaire to explain the objective of the study and sections of the instrument that the commuters will answer. The introduction instructed commuters to leave the questionnaire blank if they are not interested. A response rate of 94% was achieved, 396 completed compared with 419 collected responses.

In the questions, we incorporated the alternative responses “I do not know” or “undecided”, instead of reporting to tackle satisfying behavior of respondents in biased information retrieval [14]. In should be noted that undecided answers were considered as a middle response in the analysis [15].

In the last part of the instrument, on the same scale, the commuters were asked about the impacts of delay on various aspects of the transport, e.g., satisfaction or quality due to PIS (see Fig. 1). We hypothesized that the feelings might be associated with aspects of the PIS, which might be impacted by delay.

Fig. 1

List of all questions included in part B and C, and considered questions in part D of the instrument 

In section B, C, see Fig. 1, the respondents were asked on 5-scale Likert questions about their feelings that they might experience while facing delay. The initial examination of the data reveals that, as expected, an overwhelming majority of respondents rated the impact of delay very negatively, and in favor of various emotional or physical feelings. As discussed, part D of the instrument, which is used for the response of the analysis, contained questions about various aspects of the rail transport, e.g., quality and satisfaction.

3 Method

The three-stage least square (3SLS) refers to a method combining a system of equations being referred to SUR. The 3SLS could be seen as an extension of the two-stage least square (2SLS), while accounting for the correlation across the error terms. It is also a form of model allowing for correlations across the unobserved error terms of several equations and the instrumental variables.

The instrumental variable (IV) could be implemented by the 2SLS, or 3SLS. The application of a system of simultaneous equations is justified when there are correlations across models’ disturbance or error terms. Ignoring the possible correlation could result in biased or even erroneous results.

Another advantage of the 2SLS or 3SLS is they could account for endogeneity where one model contains a variable as a response for one equation, and as a predictor for another. Not accounting for that impact might worsen the parameters’ estimates.

A standard system of equations with no correlation across the error terms, in a system of linear equations, could be written as:

$${y}_{i}={X}_{i}{\beta }_{i}+{\varepsilon }_{i}$$

(1)

where ${y}_{i}$ is a vector of dependent variable, ${X}_{i}$ includes exogenous variables matrix, ${\beta }_{i}$ is the coefficient vector to be estimated, and ${\varepsilon }_{i}$ is the error terms. In the standard regression model, $E\left[{u}_{i}{u}_{j}\right]=0$, while we could allow for the correlation to be flexible as $E\left[{u}_{i}{u}_{j}\right]={\sigma }_{ij}$, where i and j are equation numbers, and the covariance matrix of the whole error terms could be written as $E\left[u{u}^{T}\right]=\Omega$, where $\Omega$ is Kronecker product of ${\sigma }_{ij}$, where ${\sigma }_{ij}$ is the error term across two equations for each observation. The identified matrix has the number of rows as the number of observations. Interested readers are refered to Appendix for more information regarding the model’s parameters estimations.

4 Results

4.1 Summary statistics

For the analysis, initially to check the association between feelings and various characteristics of PIS, the total of 4 psychological and 14 physiological feelings and demographic characteristics were considered. All questions except for the demographic section were based on the Likert scale. For more description about the denotation of the variables, see Fig. 1 and the related description.

As can be noticed from the response, the questions with responses of D1, D2, and D3, signal the same ideas, e.g., all about PIS, see Table 1. So, we expected the equations’ error terms to be correlated. As a result, it is reasonable to consider their correlations and analyze all the equation in the same matrices’ formulation to account for limitations of ignoring correlations.

Table 1 Descriptive summary of important factors and response

4.2 Identified parameters

For all equations, possible interaction terms were considered, and the meaningful interactions were kept in the model. The models is presented in order of Table 2: first the models, which considered various predictors for each equation, and then models with similar predictors. The main objective of conducting two sets of models was to highlight the difference of models in terms of correlation matrices across the error terms by considering similar or different predictors for equations, which are discussed later.

Table 2 Results of two considered models

For the first equation of the 1st model with D1 as response, variables of A1 and related interaction terms between B1 and B3 were highlighted to be associated with an increase in the perceived waiting time due to imprecise real-time information. The result highlight that the female commuters, compared with their counterpart (male as a reference), are less likely to believe that an inaccurate real-time information is associated with the increased waiting time. The results might be justified from the previous study’s finding that women are generally having a less sense of control, compared with their counterpart [16].

On the other hand, the interaction between B1 (angry) and B3 (frustrated) was found to be associated with an increased sense of waiting time due to the imprecise PIS. The association between feeling frustrated and increased waiting time due to imprecise PIS varies based on the level of anger that the commuters experience due to the transport delay. In other words, increased feelings of anger increase the association slope between feeling frustrated and perceived waiting time. This highlights the importance of accommodating an accurate PIS for those groups. The feelings of frustration and anger were investigated in the past study in response to service failure [17].

D2 is the second equation resulted in an increased satisfaction due to accurate real-time information as the response. The results highlight that there are associations between C10, back pain, and interaction terms of commuters who use the transport more than 8 times a week, or very regularly, and those who commute with the purpose of studying. The results highlight that there is a positive association between commuters experiencing backpain, and those who agree with an increased satisfaction due to precise performance of the real-time information.

Although there is a lack of investigating across the considered variables, the relationship between backpain and feelings stress have been extensively investigated in the literature [18]. Considering the interaction term, the association between commuters travelling with an objective of studying and perceived waiting time increases with a higher degree for commuters traveling with the purpose of studying.

The third equation is D3, which considered improved quality of the rail transport due to accurate real-time information. First, there is a positive association between B1 (angry), and the response. In other words, commuters who feel more anger due to delay are more likely to believe that the accurate real-time information has a positive impact on the quality of the transport, highlighting the importance of accommodating an accurate PIS for this group.

On the other hand, the association between the quality and interaction between commuters aged over 25 years old, and marital status was found to be important. The commuters who were married and aged over 25 are those commuters assigning a higher importance to the real-time information. That interaction is expected as the older commuters are likelier to be married. In general, the importance of the endorsement related to various aspects of the PIS increases with an increase of endorsement of various negative feelings that the commuters experience.

Moving to the 2nd model, similar variables as the first model, A1 + B3 × B1, were considered across all equations. The signs are consistent across all equations. However, while A1, for instance, is different from zero for the first equation, it is not (p-value > 0.05) for the last two equations.

4.3 Variables

Various predictors were used to reshape the predictors. Those include a combination of B1, B2, B3, A1 and A2. The results of the HT highlighted that the application of the instrument is stable and result in model improvement compared with OLS.

Although our results highlighted that the interaction across endogenous and exogenous variables could result in an improvement in the model fit through instrument variables, the second moment result in a decrease of the model fit. That might be due to the nature of our dataset being based on Likert-scale dataset.

We used few instruments for point estimates of our parameters. As highlighted in the literature, the use of many weak instruments could improve certain point estimates [19]. That is true even if each individual weak instrument is weakly correlated with the endogenous explanatory variables. So even in case of having some weak instruments, the current instrument is likely to result in model fit enhancement. The instrumental variables being considered are identical across all equations. The instrument is a combination of few endogenous and exogenous variables as B1 × B2 + B3 × D2 + A1 + A2 × D3.

4.4 Hausman test

Few checks must be made to make sure, first the instrument is not weak, and second, the 3SLS estimator is consistent. As highlighted by [20], the 3SLS estimator is consistent only and if only all disturbance terms, and all instrumental variables are uncorrelated. One of the requirements is the disturbance terms should not be correlated with the IV, which could be written by the corresponding equation ($E\left({u}_{i}^{T}{Z}_{i}\right)=0$).

The null hypothesis of the test is both 2SLS and 3SLS estimator are consistent, but the 3SLS is more efficient. Thus, if the p-value is greater than 0.05, we cannot reject the null hypothesis, and the 3SLS could be considered as a better fit. A comparison also could be made against the OLS to see if the IV is needed to improve the model fit. Thus, we did the comparisons twice. The Hausman test derives a quadratic form based on ${\chi }^{2}$, with degree of freedom equal to the number of regressors.

First the implemented method was used to compare the 2SLS, and the standard OLS with no instruments, highlighting the superiority of 2SLS, Hausman value (HV) of -28.169, with degree of freedom (DF) equal to 15, and p-value of almost 1. Thus, the OLS with no IV yields biased estimates, and the IV results in an improvement in model fit compared with a standard OLS.

The second comparison was made across the 2SLS and 3SLS. The Hausman test statistics is -13, having a chi-square distribution with 26 DF under the null hypothesis. The corresponding p-value is almost 1, highlighting that the null hypothesis cannot be rejected. Thus, the results highlight that the 3SLS estimator is asymptotically more efficient compared with 2SLS.

It should also be noted that we found the choice of instrumental variable has a significant impact on the significance of the parameters. In other words, considering just a single variable or other combination of variables as IV resulted in point estimates being not different than 0, p-value > 0.05. That is also acknowledged in the literature review [21].

4.5 Error terms’ correlation

As we discussed, it is important to take into consideration the impacts of considering similar or different predictor across various equations. As can be seen from Table 3, while considering various predictors, only slight correlation with a maximum amount across D2 and D1 (R = 0.2) could be observed. However, the highest value of correlation changes for the second model to R = 0.8. That is expected and intuitive that considering similar predictor across similar response result in higher correlation values.

Table 3 Covariance matrix of residuals

Beside the theoretical requirement of the 3SLS instrument which needs to be met, there must be enough correlating across the error terms. The results of Table 3 highlight that there are none-zero correlation across all the pairwise residuals, justifying the use of 3SLS.

5 Discussion

The main goal of this study was to investigate the association between various negative feelings that commuters experience due to delay and perceived importance of various aspects of the PIS. The case study implemented in this study was one of the most notorious rail transports that experienced occasional delay greater than an hour. Very limited studies conducted, evaluating the imprecise PIS on various commuters’ emotions, especially in a unified modeling framework and while considering their interactive relationship.

The estimation method incorporated the error structures across all equations to provide more efficient estimates, and to address the shortcomings of the standard OLS model. The comparison across few possible models highlights that the 3SLS, with instrumental variables, is more efficient than the OLS and 2SLS. The generated correlation matrices have highlighted that although the multicollinearity is not a serious issue across the residuals, when using different predictors for equations, it is a serious concern for similar predictors for all equations.

It should be noted that we consider the synthetic instrument, which is a combination of endogenous and exogenous variables, to make sure even in case of using few weak instruments, the combination of those weak instruments result in a reliable instrument. The test was used to make sure that the instruments result in the robustness of the model. The test on the strength of the instrument, by testing the 3SLS method against OLS revealed that the instrument is not weak and results in a superiority of the 3SLS model.

In summary, we found that gender, being angry, frustrated, the objective of travel, and the frequency of using the transport are some of the factors associated with various aspects of real-time information.

Several important findings could be derived from the present study. Possibly, the most important finding is the strong and significant associations across various feelings that the commuters experience and perceived quality of real-time information. In other words, the commuters who experience a higher level of negative feelings are expected to assign a higher importance to the accuracy of real-time information, and its impact on the perceived quality of a rail transport. Association was also found between demographic characteristics of commuters and real-time information. The policy could be adopted to offset the costs of delay on commuters by providing accurate information for the commuters so the negative feelings would be alleviated.

The results of interaction terms highlighted that the relationships between many variables and responses are not stable but vary based on other predictors. For instance, it was found that the relationship between more satisfaction due to the accurate real-time information and purpose of travel varies based on the frequency that the commuters use the transport service. Also, the increase in the perceived quality of the rail transport due to accurate PIS and age varies based on the frequency that the commuters use the service.

The results generally point to the higher importance of the real-time information for those commuters who use the rail transport more frequently than others. Thus, special attention needs to be given to that group. Even if the policy makers are not able to provide a reliable and timely transport service, we recommend that they work to provide precise real-time information so commuters can adjust their expectation based on the provided information.

Also, the association between feeling frustrated and delay, and the belief that imprecise real-time information result in an increase in perceived waiting time varies based on feeling angry. In general, commuters who feel more anger and frustration experience increased waiting time due to imprecise real-time information. The result could be linked to the negative feelings causing individual to experiencing time as moving more slowly [22].

The interaction between age and frequency of using the transport highlight that the association between age and increased perceived quality of the transport due to accurate real-time information varies based on frequency of using the transport. The results highlight that the younger commuters, in general, more strongly agree that accurate real-time information impacts the quality of the transport more positively. That might be a sign that those groups require the transport more frequently for reaching their destination mainly due to the possible purpose of education. Also, those groups are more likely to take advantage of the real-time information.

Few arguments could be set forward regarding the limitations of this study. First, this study assumed that each passenger’s journey was delayed, and it was delayed always by the same amount. As a result, if a passenger did not experience a real delay at the time of the survey, their responses would differ from other responses. In response to that concern, beside the tremendous shortcoming of the rail transport in terms of its delay, the questions were, specifically, asked in a general sense and not specific to the time of survey. Future studies should take into account the complex association of factors and delay while modeling various aspects of a rail transport.

Appendix

1.1 Model parameters estimations

The point estimates of the 3SLS could be written as:

$${\widehat{\beta }}={({\widehat{X}}^{T}{\widehat{\Omega }}^{-1}\widehat{X})}^{-1}{\widehat{X}}^{T}{\widehat{\Omega }}^{-1}y$$

(2)

where all items were defined before. The process of the above equation would be estimated efficiently by estimating common parameter of ${{\widehat{X}}^{T}{\widehat{\Omega }}^{-1}}$ and through ${X}^{T}{\Omega }^{-1}=\sum{\left[\begin{array}{c}{\sigma }^{1i}{X}^{1}\\ \begin{array}{c}{\sigma }^{2i}{X}^{3}\\ \begin{array}{c}.\\ {\sigma }^{3i}{X}^{3}\end{array}\end{array}\end{array}\right]}^{T}$[23]. It should be noted that the Eq. 2 is an extension of a simple linear regression coefficient estimate of Eq. 1, which follows:

$${\widehat{\beta }}={({\widehat{X}}^{T}\widehat{X})}^{-1}{\widehat{X}}^{T}y$$

(3)

Equation 3 resulted from minimization of the sum of squared of residuals $\sum {\varepsilon }_{2}^{2}$.

The process of the 3SLS is simple and follows some mathematical equations. The following paragraphs outline the simplified process. To fulfill the objectives of accounting for correlation across the error terms, it is necessary to analyze a system of equations simultaneously, while identifying the contributory factors in those equations.

The process of the method, like any algorithm, consist of an objective and cost functions. Cost function is the residual, while the objective function is minimization of that cost function. Another point is to account for the correlation across the equations’ nuisances. To achieve that, matrices would be made to account for various equations at the same time. So, the correlations between the equations could be accounted for by considering covariance matrices of $\beta$ s. The covariance could be written as:

$$\widehat{COV}[\widehat{\beta }]={({X}^{T}{\Omega }^{-1}X)}^{-1}$$

(4)

As discussed, $\Omega$ is based on $\Sigma$, and $\Sigma$ accommodates all${\sigma }_{ij}$. X, on the other hand, accommodate the instrumental variables as follows:

$${\widehat{X}}_{i}={Z}_{i}{({Z}_{i}^{T}{Z}_{i})}^{-1}{Z}_{i}^{T}{X}_{i}$$

(5)

where ${\widehat{X}}_{i},$ in case of presence of instrumental variables, would be estimated based on a single or combination of instruments’ variables of ${Z}_{i}$. As discussed, ${\widehat{\beta }}$ follows Eq. 2 as${\widehat{\beta }}={({\widehat{X}}^{T}{\widehat{\Omega }}^{-1}\widehat{X})}^{-1}{\widehat{X}}^{T}{\widehat{\Omega }}^{-1}y$: where ${\widehat{\beta }}$ matrix stacks all the parameters, and consequently the optimization process would be employed.

In summary, there is y, or response, which accommodates responses across all equations, and the same thing for predictors’ vectors across the equations. Z matrix is made based on predictors and is aggregated based on sets of variables, which could be exogenous, endogenous, or a combination of both. That would be used to modify the X based on Eq. 5. Again, stacking would be done for y, x and z.

For estimation of ${\widehat{\sigma }}_{ij}$, the disturbances of the two equations of i and j as ${u}_{i} and {u}_{j}$ are needed, which would be estimated based on the cost function as:

$${\widehat{u}}_{j}= y-\beta *X$$

(6)

where all the parameters were defined earlier.

The covariance matrix is calculated from estimated residuals of ${\widehat{u}}_{i} and {\widehat{u}}_{j}$ based on Eq. 7, as the true disturbances are not known [23]. That could be written as:

$${\widehat{\sigma }}_{ij}=\frac{{\widehat{u}}_{i}^{T}{\widehat{u}}_{j}}{{((T-{K}_{i}).(T-{K}_{j}))}^{0.5}}$$

(7)

where the numbers of parameters in equations i and j are ${K}_{i} \,and\, {K}_{j}$ respectively.

After the ${\widehat{\sigma }}_{ij}$ is created based on the above, or Eq. 7, values would be updated in a loop. In each step in a loop, the coefficient would be estimated based on ${({\widehat{X}}^{T}{\widehat{\Omega }}^{-1}\widehat{X})}^{-1}{\widehat{X}}^{T}{\widehat{\Omega }}^{-1}y$, which incorporate the updated residuals, $\Omega$. The process would be iterated until reaching the tolerance for residuals.

In summary, the goal is to estimate the parameters in single matrices so the correlation could be considered across the disturbances. It is clear that the p-value would be estimated from standard error, and standard error through square root of coefficients covariance. The covariance matrix would be created in R environment by solving the linear systems of equations of ${\left({\widehat{X}}^{T}{\widehat{\Omega }}^{-1}\widehat{X}\right)}^{-1}{\widehat{X}}^{T}{\widehat{\Omega }}^{-1}y.$

The Hausman test is used to check the appropriateness of the instrument. It compares the finalist models with one of the other alternatives. To check the suitability of the IV, we would check it against the ordinary least square (OLS). Similar to the previous study [24], which used Hausman test on OLS and 3SS, we would also use the technique to determine if the results of OLS are inconsistent, and thus determine whether the inclusion of the IV is appropriate or not.

The check also would be made across 2SLS and 3SLS, with the same method. The Hausman test (HT) could be written as:

$$HT={\left({\widehat{\beta }}_{h}-{\widehat{\beta }}_{3SLS}\right)}^{\intercal }(\widehat{COV}\left[{\widehat{\beta }}_{h}\right]-\widehat{COV}\left[{\widehat{\beta }}_{3SLS}\right])({\widehat{\beta }}_{h}-{\widehat{\beta }}_{3SLS})$$

(8)

where $\widehat{COV}$ is the estimated coefficient, and ${\widehat{\beta }}_{h}and {\widehat{\beta }}_{3SLS}$ are hypothesis estimated coefficient and the coefficient related to 3SLS, respectively.