All articles by Rashmi Sajwan

Establishing a relationship between FDI and air pollution in India

The previous articles in this study established how Foreign Direct Investment (FDI) affects the economic growth of a country. In India, in the last two decades, the inflow of FDI has grown significantly. Similarly, its environmental pollution has also been rising since 1991 due to an increase in economic activity. This article empirically investigates the impact of FDI on air pollution in India. In order to do so, it establishes a long run association between FDI inflows and air pollution in the post reform period, i.e. since 1991. The previous article identified the common air pollution indicators, of which Greenhouse Gas (GHG) emissions is the most significant. The figure below shows that total GHG emissions have increased drastically in India. Read more »

Understanding normality test in STATA

The preceding articles showed how to conduct time series analysis in STATA on a range of univariate and multivariate models including ARIMA, VAR (Lag selection and stationarity in VAR with three variables in STATA) and VECM (VECM in STATA for two cointegrating equations). Time series data requires some diagnostic tests in order to check the properties of the independent variables. This is called ‘normality’. This article explains how to perform normality test in STATA. Read more »

How to test time series multicollinearity in STATA?

After performing autocorrelation tests in STATA in the previous article, this article will explain the steps for detecting multicollinearity in time series. The problem of multicollinearity arises when one explanatory variable in a multiple regression model highly correlates with one or more than one of other explanatory variables. It is a problem because it underestimates the statistical significance of an explanatory variable (Allen, 1997). A high correlation between independent variables will result in a large standard error. This will make the corresponding regression coefficients unstable and also statistically less significant. Read more »

How to test time series autocorrelation in STATA?

The previous article showed how to perform heteroscedasticity tests of time series data in STATA. It also showed how to apply a correction for heteroscedasticity so as not to violate Ordinary Least Squares (OLS) assumption of constant variance of errors.  This article shows a testing serial correlation of errors or time series autocorrelation in STATA. Autocorrelation problem arises when error terms in a regression model correlate over time or are dependent on each other. Read more »

How to perform Granger causality test in STATA?

A previous article (Lag selection and cointegration test in VAR with two variables) in this module demonstrated the application of cointegration test in time series analysis. Applying Granger causality test in addition to cointegration test like Vector Autoregression (VAR) helps detect the direction of causality. It also helps to identify which variable acts as a determining factor for another variable. This article shows how to apply Granger causality test in STATA. Read more »

How to perform Heteroscedasticity test in STATA for time series data?

The previous articles showed how to perform normality tests in time series data. This article focuses on another important diagnostic test, i.e. heteroscedasticity test in STATA. Heteroskedastic means “differing variance” which comes from the Greek word “hetero” (‘different’) and “skedasis” (‘dispersion’). It refers to the variance of the error terms in a regression model in an independent variable. Read more »

Slacks based measure or SBM analysis in DEA

Slacks based measure (SBM) is a non-radial model to solve the problem in the “additive model” developed by Charnes, Cooper, & Rhodes in 1978. This model can discriminate between efficient and inefficient Decision-Making Units (DMU). However it does not provide an efficiency measure, so a decision maker cannot interpret the performance of the DMU. Therefore “super SBM” model was introduced to determine and rank efficient DMU. The efficiency scores are dimensionless and fall between 0 and 1, thus allowing for the comparison of different DMU in terms of efficiency.

Why SBM model?

SBM model uses “slacks” to show excess input and shortfalls in the output then it directly deals with them by maximizing these slacks. SBM analysis provides an efficiency score which is units-invariant and a monotone function of input slacks and output slacks. To rank efficiency, super SBM was introduced by Tone & Tsutsui (2001b). It is appropriate for evaluating efficiencies when inputs and outputs may change non-proportionally.

Super SBM model can be expanded by the super efficiency to the DEA models. Super efficiency rate refers to the distance between the inputs and outputs of both units. The distance is shown in variable ρ. Assuming that there exist a set of ‘n’  DMU producing the same set of outputs which consume the same set of inputs. Input and output matrix is matrix (X, Y), where (Input) X= (xij) ϵ Rmxn and (Output) Y= (yij) ϵ Rsxn.

Equations of the SBM model

λ is a nonnegative vector in Rn. The vector S ϵ Rm and S+ ϵ Rs, shows an excess input and a short falling output, respectively.  The equation for the SBM model is as follows:

Figure 1: Equation for SBM analysis model

Figure 1: Equation for SBM model

Suppose (ρ*, λ*,s-*, s+*) is the optimal condition for SBM analysis and (x0 , y0) is SBM efficient of DMU. When ρ* = 1, s-*=0 and s+*=0 (or there is no excess input and a short falling output). A super-efficiency model helps in ranking DMU. Below is the formula for super SBM analysis:

Figure 2: Equation for SBM model

Figure 2: Equation for super SBM model

Challenges of super SBM model

One of the major challenges in super SBM model is that it involves solving a complex equation. This becomes more tedious when there are large units of input and outputs. It takes a lot of effort and time to estimate the efficiency of different decision-making units manually. Therefore, it is optimal to use statistical packages like DEA solver, max DEA, and others. They are more popular than manual calculations and other measures of evaluating efficiency.

Interpretation of super SBM analysis results

This module shows the following process in determining efficiency for 10 DMU and ranks them on the basis efficiency by using Super SBM model. It uses MaxDEA software to assess efficiency. MaxDEA 7 basic is available free of cost, however, the ‘pro’ version is a paid software.

Input and output dataset for 10 DMU




Exports percentage of GDP




Exchange rates

1 68.12 3,069,559.00 2.27 30.73
2 19.99 3,488,123.37 1.70 1.04
3 29.67 3,696,265.19 5.02 0.64
4 100.63 4,138,413.19 1.35 3.67
5 45.40 4,736,995.93 3.79 0.75
6 75.63 4,984,467.88 1.72 3.15
7 53.88 6,682,944.91 3.96 1,094.85
8 24.50 13,177,694.49 3.04 6.20
9 15.92 13,544,244.95 3.54 97.60
10 13.64 23,869,948.68 2.65 1.00

By using maxDEA software and applying super SBM model, efficiency score for all DMU can be determined  and ranked. DEA-SBM model is non-radial and non-oriented and can deal with inputs and outputs individually. The purpose is to minimize the input and output slacks, resulting in this fractional program.

Efficiency scores and ranking from super SBM model




1 0.248082 8
2 0.541691 6
3 0.651521 5
4 1 1
5 1 1
6 0.796988 4
7 0.151898 10
8 0.359254 7
9 0.227246 9
10 1 1

Determining the efficiency score of DMU

The results show that DMU4, DMU5, and DMU6 are the most efficient decision-making units achieving highest efficiency score of 1. The efficiency score of all other DMUs is less than 1 in DEA-SBM models. It reflects these DMUs have input excesses and output shortfalls. This analysis can be used in yearly data. The mean efficiency score for specific DMU over a period of time and year time efficiency can be evaluated too.


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