Robust Regression Techniques: Econometric Approach (2021) PDF

Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 6525079, 9 pages https://doi.org/10.1155/2021/6525079 Research Article Applications of Robust Regression Techniques: An Econometric Approach Dost Muhammad Khan ,1 Anum Yaqoob,1 Seema Zubair,2 Muhammad Azam Khan,3...

Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 6525079, 9 pages https://doi.org/10.1155/2021/6525079 Research Article Applications of Robust Regression Techniques: An Econometric Approach Dost Muhammad Khan ,1 Anum Yaqoob,1 Seema Zubair,2 Muhammad Azam Khan,3 Zubair Ahmad ,4 and Osama Abdulaziz Alamri5 1 Department of Statistics, Abdul Wali Khan University, Mardan, Pakistan 2 Department of Statistics, Mathematics and Computer Science, The University of Agriculture, Peshawar, Pakistan 3 Department of Economics, Abdul Wali Khan University, Mardan, Pakistan 4 Department of Statistics, Yazd University, P.O. Box 89175-741, Yazd, Iran 5 Statistics Department, University of Tabuk, Tabuk, Saudi Arabia Correspondence should be addressed to Zubair Ahmad; [email protected] Received 14 April 2021; Accepted 21 May 2021; Published 29 May 2021 Academic Editor: Amer Al-Omari Copyright © 2021 Dost Muhammad Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Consistent estimation techniques need to be implemented to obtain robust empirical outcomes which help policymakers formulating public policies. Therefore, we implement the least squares (LS) and the high breakdown robust least trimmed squares (LTS) regression techniques, while using econometric regression model based on a growth equation for the two countries, namely, India and Pakistan. We used secondary annual time series data which covers a long period of 41 years. The adequacy of the time series econometric model was checked through cointegration analysis and found that there is no spurious regression. Classical and robust procedures were employed for the estimation of the parameters. The empirical results reveal that the overall fit of the model improves in case of LTS technique, while the significance of the predictors changes significantly in cases of both countries due to the removal of outliers from the data. Thus, empirical findings exhibit that the results, obtained through LTS, are better than LS techniques. 1. Introduction Several prior studies have been carried out to investigate the relationship of economic growth measured by GDP with The rise in the productive capacity of goods and services over FDI inflows, gross savings, and population growth. How- certain period of time of a particular economy is termed the ever, there are conflicting views regarding the impact of FDI economic growth. Conventionally, it is measured in terms of and population growth on economic growth, e.g., analyzing gross domestic product (GDP). However, real GDP is the the relationship between FDI and economic growth of India. most commonly used measure of the economic growth of a For example, Ray found a positive relationship between country’s economy in the literature. A rise in the GDP of a the two. Keshava also found a positive relationship, but nation is referred to as economic growth which changes the relatively insignificant impact of FDI on GDP of India. yearly with the recession and expansion of the economy. Saqib et.al. (2013) investigated the impact of FDI along with Economic growth is one of the key aims of development four other variables, debt, trade, inflation, and domestic policy in almost every country. The economic growth ac- investment, on the GDP of Pakistan for the period of celerates society in a positive and productive direction and is 1981–2010. Their findings indicate that FDI, debt, trade, and influenced by a variety of factors, including human capital, inflation hurt the GDP while domestic investment has a physical capital, institutional factors, interest rate, exports, positive impact. Zeb et al. studied the impact of FDI government expenditures, population growth, savings, and along with three other variables: trade openness, political inflation [1, 2]. instability, and terrorist attacks on the economic growth of 2 Mathematical Problems in Engineering Pakistan over the period of 1972–2012. They applied the least 1975 to 2015. The robust regression technique encompasses square method to investigate the impact of these variables on the least squares (LS) method and the robust LTS technique the GDP of Pakistan. The results unveil the positive and. The simultaneous use of both techniques is expected to significant effect of FDI on the economic growth of Pakistan. highlight the key differences in the development paths of Likewise, Mehmood explored the impact of thirteen both countries comparing the output of both countries. The selected factors, including FDI inflow and gross saving on chosen set of variables is FDI, gross savings and population the GDP of Pakistan and Bangladesh. His findings are gross growth, and real GDP per capita. To the best of the authors’ saving that has a positive and significant impact on the GDP knowledge, this is a pioneer study on the application of of Pakistan, whereas FDI is the insignificant indicator of robust regression methods using data from the two devel- GDP. A review of related literature shows that the mixed oping Asian countries. The main contribution of this article impact of FDI on the economic growth depends on several is to capitalize on the weaknesses of the classical estimation factors, such as domestic infrastructure, highly educated methods in the presence of outlying observations in the data labor force, trade regime, and FDI policies of the host that cause ill-estimation by providing alternative methods country. Azam and Ahmed found that FDI inflow that are robust and less sensitive towards outliers and en- contributes to the economic growth in ten Commonwealth compass the weaknesses of the classical estimation methods. of Independent States during 1993–2011. The rest of the paper is organized as follows: Section 2 Both Pakistan and India are facing the same problem of deals with empirical methodology covering regression rapid population growth. According to the World Pop- analysis, outliers’ management, robust regression, robust ulation Data Sheet (2016), Pakistan is the sixth most pop- regression through least trimmed squares (LTS), and data ulous country of the world, whereas India is the second-most description. Section 3 presents spurious regression and populous country. To assess the impact of population growth cointegration analysis, Section 4 includes model specifica- on the economic growth of a nation, some researchers find a tion, and Section 5 presents empirical findings. Finally, negative impact of the population growth that gives rise to Section 6 discusses the conclusion of the study. the unemployment problem, lack of health and educational facilities, and reduction in the household savings which in 2. Empirical Methodology and Data turn lowers the national savings, while others argue that the high population growth is the real power of a nation as it This section includes the description of the data collected, gives rise to high labor force which helps the country by the variables under consideration, the appropriate models giving high output. For example, Afzal discovered the used in this study, and statistical analysis for its estimation negative impact of population growth on economic growth, that have been carried out to meet the desired objectives. whereas Ali et al. noted the positive impact of pop- ulation growth on the economic growth of Pakistan, which contradicts the findings made by Afzal. Kothare 2.1. Data Sources. As mentioned earlier, the time series data argues that India is the fastest-growing economy in the on the selected variables are taken from World Development world because of its rapid population growth which has a Indicators published by the World Bank for the period positive impact on the GDP of the country. Similarly, of 41 years from 1975 to 2015. The variables that are con- Koduru and Tatavarthi highlighted the positive impact sidered here for this study are FDI inflow, annual population of population growth on the economic growth of India. growth, gross savings, and GDP per capita of Pakistan and Using the pooled mean group approach, Olayungbo and India. To make the interpretation simpler, the data are Quadri found that output per labor, population, trade transformed by log transformation. openness, remittances, and FDI have a positive impact on economic growth in 20 sub-Saharan African countries 2.2. Regression analysis. Regression analysis is one of the during 2000–2015, while the inflation rate has a negative important and commonly used statistical tools for investi- impact on growth. Azam et al. observed that population gating the relationship between a dependent and one or growth, life expectancy, and investment have significant and more independent variables, with wide applications in the positive impacts on economic growth in India from 1980 to field of finance, economics, medicine, and psychology. A 2018, whereas the inflation rate variable has a negative link regression model is generally defined as with economic growth. Azam and Feng found that official development assistance and inflation rate had neg- Y Xβ + ε, (1) ative, while FDI inflows, trade, and human capital by gross secondary school enrollment (%) had positive impacts on where the dependent variable Y and the vector of true re- economic growth for 37 developing countries over 1985 to siduals ε are n × 1 and the design matrix X is n × p. Write β􏽢 2018. for an estimate of β, and In their study, Zaman et al. claim that “a literature 􏽢 e Y − Xβ, (2) search shows that robust regression techniques are rarely used in applied econometrics.” Therefore, the central pur- for the corresponding fitted residuals. pose of this study is to implement robust regression tech- The regression analysis commonly makes use of the least niques, while using data from two developing countries, squares method for estimation of model parameters under namely, India and Pakistan, over the period ranging from some assumptions to be satisfied, such as the normality of Mathematical Problems in Engineering 3 errors with zero mean and constant variance, i.e., ε ∼ N (0, breakdown point increases. For example, OLS has a breakdown δ2). The least square principle is to estimate the parameters point of 0% which represents that even a single outlier is by minimizing the sum of squared residuals (difference sufficient to distort the OLS estimators. The robust techniques between the actual and the fitted values of the dependent have 50% of breakdown point which is considered as the variable). Thus, the least squares (LS) method uses the highest breakdown point. The property of bounded influence function ρ(e) e2 , which is extremely sensitive to outliers, measures the resistance of the estimator against bad obser- particularly those occurring on high leverage cases. The least vations. It encounters the tendency of the least squares to allow square method gives misleading results when the assump- the leverage points to exhibit greater influence. tion of normality is dissatisfied or outliers happen in the data There are various robust regression techniques. The first as outliers drag the least square fit towards itself. Because of step in this respect came from Edgeworth who proposed the extreme sensitivity of least square, a single outlier in a the least absolute deviation (LAD) by minimizing the sum of large sample is sufficient to deviate the regression fit totally absolute residuals instead of minimizing the sum of squared as its breakdown point is 1/n which tends to zero with the residuals. LAD is preferable over OLS in providing protection increase in sample size n. against vertical outliers but is worse in the case of high le- verage points with a breakdown point of 1/n. Another popular approach is based on Huber’s M-estimator which 2.2.1. Outliers Management. Outliers being inconsistent minimizes a symmetrical objective function of residuals in- observations and largely deviated from the majority of the stead of squared residuals. M-estimator is robust against observations in data need proper handling as they pose outliers in location and is more efficient than LAD. There are serious threat to the regression model and its estimated some other robust procedures, including least median square coefficients and, as a result, give misleading outcomes. Two by Rousseeuw , which minimize the median of squared types of outliers can happen in the regression dataset. One residuals, having 50% breakdown point, but low efficiency. To with extremely large values in the response is referred to as overcome the issue of low efficiency and to maintain high vertical outliers, whereas observations with extremely large robustness, Rousseeuw introduced least trimmed square values in the explanatory variable are called leverage points. estimators (LTS) which minimize sum of the smallest h Outliers being influential to the classical regression require [(n + p + 1)/2] squared residuals. Other robust estimators methods that are insensitive to it. Two approaches are include S-estimator and MM-estimators.. commonly used to cope with this problem, popularly known as diagnostic approach and robust procedures. The diagnostic approaches try to identify the unusual 2.2.3. Robust Regression through Least Trimmed Squares observations through diagnostic statistics and remove it (LTS). Least trimmed squares (LTS) is a highly robust and from the data and then classical procedures, for example, comparatively efficient estimator among all the robust es- least square estimation procedure is then applied to the timators available in the literature and is obtained by remaining clean dataset. This approach is suitable for simple minimizing the trimmed sum of the squared residuals. data or when there are one or two outliers, but it becomes LTS is a modified form of the LS estimator which corre- inefficient to detect outliers in case of multiple outliers sponds to the more central values by ignoring the extreme present in a multidimensional dataset. Therefore, an ap- observations in the ordered data. Consider the model in propriate procedure to deal with outliers is robust proce- equation (1): dures that not only detect multiple outliers in complex data but also give efficient results. Y Xβ + ε. (3) High breakdown admits the possibility that a large 2.2.2. Robust Regression. A robust regression is an iterative fraction of the data may have been replaced by arbitrary procedure that is designed to overcome the problem of outliers values. The high breakdown approach is exemplified by the and influential observations in the data and minimize their least trimmed squares (LTS) criterion. Here, we write impact over the regression coefficients. Most of the re- gression techniques, named as robust, do not have this 􏽢 e1 y1 − x1 β. (4) property. The main objective of robust estimation is to obtain reliable estimates/inferences for unknown parameters in the Defining ei as the corresponding residual, write e(i) for presence of outliers. The robust procedure replaces the sum of the ith order statistic of the ei. That is, 􏼌􏼌 􏼌􏼌 􏼌􏼌 􏼌􏼌 􏼌􏼌 􏼌􏼌 squared residuals of the OLS with some other function that is 􏼌􏼌e(1) 􏼌􏼌 ≤ 􏼌􏼌e(2) 􏼌􏼌 ≤... ≤ 􏼌􏼌e(n) 􏼌􏼌. (5) being less influenced by the unusual observations. These procedures first fit a regression to the data and then identify the Then, LTS is defined by the criterion outliers as those observations having large residuals. Robust h techniques have three desiring properties, namely, efficiency, 2 β􏽢 min 􏽘 e(i) , (6) breakdown point, and bounded influence. The breakdown i 1 point is the smallest fraction of the unusual observations that an estimator can tolerate before giving an incorrect result. It is where h is a coverage parameter, commonly chosen to be always a value between zero and 0.5. It measures the degree of h [(n + p + 1)/2] that determine the robustness of LTS. robustness, the robustness of an estimator increases as the With the choice h n, LTS specializes in OLS. 4 Mathematical Problems in Engineering LTS is more efficient than least median squares (LMS), Since the variables at first difference become stationary but its computational procedure is more complicated as as given in Table 1, so the variables are cointegrated and compared to LMS. It has a greater convergence rate [24, 25]. there is no spurious or nonsense regression. Therefore, all The LTS procedure first identifies the outliers as the points these variables can be used in the multiple regression model with extreme positive or negative residuals. It then proceeds defined in (7). with the OLS for improved accuracy to classify a data point as outlier, standardized residuals are calculated, and data 4. Model Specification point with standardized LTS residual in its absolute value greater than 2.5 is considered as outlier. It helps in detecting The multivariate regression model within the framework on the outliers efficiently. This goal is hard to achieve, otherwise, economic growth equation which is also used by many prior in high dimension data. It may be possible for LTS to detect studies including the studies by Hasan , Adenola and too many data points as outliers, but removing a large Saibu , Peter and Bakari , and Azam [31, 32] and to proportion of data points as outlier may result in a re- be used for the two datasets is expressed as follows: gression that does not completely reflect the desired rela- tionship. In their study, Roozbeh and Arashi noted Log GDPt β0 + β1 Log FDIt + β2 Log PGt + β3 Log GSt + εt , that the LTS estimator is a highly robust regression esti- (8) mator, while it is well known that the method of least squares is very sensitive to outliers. where LogGDP is the gross domestic product per capita, LogFDI is the foreign direct investment, net inflows (% of GDP), LogPG is the population growth (annual %), and 3. Spurious Regression and LogGS is the gross savings (% GDP). The GDP is specified as Cointegration Analysis the dependent variable and the remaining three variables are the explanatory variables and ε is the error term. The GDP The concept of cointegration introduced by Granger has per capita (constant 2010 US $) has been used as a proxy for turn out to be extremely important in the analysis of economic growth. The data are in log form to avoid any nonstationary economic time series. To illustrate this nonlinearity problem in the data. problem, consider a simple regression model Y t β 0 + β1 X t + εt , (7) 5. Empirical Results and Discussion where Yt is a dependent variable, Xt is a single independent To apply the robust regression techniques on the two regressor, and εt is a white noise term with mean zero developing Asian countries, namely, Pakistan and India, sequences. the classical least squares and robust LTS regressions are If both Xt ∼I (1) and Yt ∼I (1), by differencing one time, it being used. Since the data used in the model are time series is called integrated Yt and Xt of order 1, denoted by I (1). and the error terms of time series data often suffer from Then, generally Yt − βXt ∼ I(1) as well. However, there is autocorrelation, therefore, Newey–West HAC (hetero- one important exception. If εt ∼ I(0), then Yt − βXt ∼ I(0), scedasticity and autocorrelation consistent) estimation is that is, the linear combination of Yt and Xt has the same considered for correcting the OLS standard errors in case of statistical properties as I (0) variable. In this case, variables unknown autocorrelation and heteroscedasticity of the Yt and Xt are called cointegrated. errors. The Newey–West estimation procedure gives the From regression models, the observed value of the t-sta- same estimates of the regression parameters as the OLS, but tistic of the coefficient estimates is calculated under the as- different values of the standard errors result in different sumption that the true value of the coefficient is equal to zero; t-statistic and p value for testing the null hypothesis. despite this fact, researchers have found that the null hypothesis Moreover, HAC estimation is valid in the case of large (H0) of zero was rejected much more frequently than standard samples and gives better results than OLS, since the sample theory predictions. These results indicated that many of the size of 41 observations used in this analysis is reasonably significant relationships between nonstationary economic large; therefore, HAC estimation procedure is imple- variables and existing econometric models will be spurious. mented here. The regression estimates of both LS and LTS Researchers dealing with time series variables often are presented in Tables 2 and 3. It is evident from Table 2 suggest a simple solution to the problem of spurious re- that the LS results suggest that population growth affects gression. If the relationships between economic variables are economic growth negatively and significantly, i.e., a rise in specified in the first differences, the analytical complications the population growth is associated with the decrease in due to nonstationary variables can be avoided because the economic growth. Results also reveal that the FDI inflow differenced variables become stationary even if the original and gross savings have a positive, but insignificant impact variables are not. on the GDP of Pakistan. The overall regression model is From Figures 1 and 2, it can be seen that there is highly significant. The predictors explain 79.55% of the nonstationary stochastic trend in all the variables of the variation in the GDP per capita. Figure 3 shows the India and Pakistan GDP datasets. Thus, augmented Dick- standardized residuals versus fits in the case of Pakistan, ey–Fuller (ADF) unit root test for testing the null hypothesis whereas Figure 4 shows the standardized residuals versus of nonstationarity was applied. robust distance for Pakistan data. Mathematical Problems in Engineering 5 7.5 3.6 Log gross saving Log GDP 7.0 6.5 3.2 6.0 2.8 1980 1990 2000 2010 1980 1990 2000 2010 Year Year (a) (b) Population growth FDI (% of GDP) 3 2.0 2 1.6 1 0 1.2 1980 1990 2000 2010 1980 1990 2000 2010 Year Year (c) (d) Figure 1: Time series plot of (a) log of GDP, (b) log of gross saving, (c) foreign direct investment, and (d) population growth for Indian data. 7.0 3.4 Log gross saving 3.2 Log GDP 6.6 3.0 2.8 6.2 1980 1990 2000 2010 1980 1990 2000 2010 Year Year (a) (b) 1 1.5 Population growth FDI (% of GDP) 0 –1 1.3 –2 1.1 1980 1990 2000 2010 1980 1990 2000 2010 Year Year (c) (d) Figure 2: Time series plot of (a) log of GDP, (b) log of gross saving, (c) foreign direct investment, and (d) population growth for Pakistan data. Table 1: Augmented Dickey–Fuller test and P values. India Pakistan Variables ADF P value ADF P value LogGDPt −6.093 0.001 −3.442 0.006 LogGSt −6.014 0.010 −4.942 0.01 LogFDIt −4.113 0.013 −5.199 0.01 LogPGt −5.083 0.010 −3.963 0.021 6 Mathematical Problems in Engineering Table 2: Regression estimates (analysis of economic growth of Pakistan). LS Robust Var. Coeff. S.E t-val. P val. Coeff. S.E t-val. P val. Const. 3.5050∗ 0.2358 14.8610 0.0000 3.2338∗ 0.1029 31.428 0.0000 Log (FDIt) 0.0409 0.0445 0.9188 0.3641 0.1044∗ 0.0173 6.038 0.0000 Log (PGt) −0.4112∗ 0.3854 3.6617 0.0008 −0.7934∗ 0.1155 6.866 0.0000 Log (GSt) 0.1286 0.1454 0.8850 0.3819 0.0770 0.0642 1.199 0.2400 R2: 0.8108 Adj. R2: 0.7955 R2: 0.9336 Adj. R2: 0.927 F-statistic: 52.87 F-statistic: 140.70 Residual standard error: 0.0489 Residual standard error: 0.0249 ∗ Note. represents the level of significance at the 1% level. Table 3: Regression estimates (analysis of economic growth of India). LS Robust Var. Coeff. S.E t-val. P val. Coeff. S.E t-val. P val. Const. 3.1756 0.0830 38.2339 0.0000 3.0939 0.0937 33.025 0.0000 Log (FDIt) 0.0086 0.0117 0.7347 0.4672 0.0055 0.0059 −0.9380 0.3548 Log (PGt) −2.0359 0.0576 −35.3675 0.0000 2.0528 0.0664 −30.895 0.0000 Log (GSt) 0.1455 0.0542 2.6856 0.0108 0.2017 0.0572 3.5250 0.0012 R2: 0.9944 Adj. R2: 0.9939 R2: 0.9958 Adj. R2: 0.9954 F-statistic: 2192 F-statistic: 2735 Residual standard error: 0.0158 Residual standard error: 0.01367 ∗ Note. represents the level of significance at the 1% level. 41 38 39 40 4 Standardized LTS residuals 37 36 2.5 2 0 –2 1 –2.5 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Fitted values Figure 3: Standardized residuals vs fitted values. Regression diagnostic plot 41 39 38 Standardized LTS residuals 4 40 2.5 2 9 15 14 12 13 10 0 16 11 8 17 2 7 6 3 4 –2 5 1 –2.5 0 1 2 3 4 5 6 7 Robust distance computed by MCD Figure 4: Standardized residuals vs robust distance. Mathematical Problems in Engineering 7 Figure 5 shows the standardized residuals versus fitted which indicates that the FDI inflow has an insignificant values for India and Figure 6 shows the standardized re- negative impact on the GDP per capita of India. Therefore, it siduals versus robust distances for India. results in a decrease of economic growth to some extent. The robust analysis indicates that the population growth Similarly, population growth lowers the GDP per capita. The and foreign direct investment inflow both contribute sig- improvement in the model fit by LTS is also indicated by the nificantly to the economic growth of Pakistan. However, FDI value of R2, F-value, and residuals standard error. inflow is positively related, and population growth is neg- Empirical findings of the present study, regarding the atively related to economic growth, whereas the impact of relationship of FDI inflow and GDP per capita, do not verify gross savings is insignificant. the Ray and Keshava’s findings, who explored the Least trimmed squares reveals that there are 8 points positive relationship between FDI and economic growth. with standardized residuals bigger than 2.5 standard devi- Similarly, the results of this study regarding the impact of ations in absolute values. These values indicate the years population growth on economic growth do not verify the 1975, 2009, 2010, 2011, 2012, 2013, 2014, and 2015 with findings of the studies conducted by Kothare and standardized LTS residuals of −3.77, 2.65, 3.01, 4.39, 5.69, Koduru and Tatavarthi who argue that population 5.33, 5.17, and 6.89, respectively, whereas the standardized growth positively affects economic growth. All these results residuals plot and regression diagnostic plot indicate that suggest that the government of India should not rely on the five points 2011, 2012, 2013, 2014, and 2015 are the influ- FDI inflow and population growth for the improvement of ential observations, and removal of these 5 points has a large the economic growth. The comparison of the outputs of both influence over the regression. The FDI inflow though in- techniques represents that least square estimates are highly significant with the classical method appears significant with affected by outliers and give significantly different results the removal of outliers. The overall fit of the model improves from that of the LTS results which are in accordance with the as the R2 and adjusted R2 substantially increase to 93.36% findings of Zaman et al. , Al-Athari and Al-Amleh , and 92.7%, respectively. and Onur and Cetin , who found that least squares The negative impact of the population growth reveals method gives invalid estimates in the presence of a single that rapid population growth is alarming and slows down outlying observation in the data, while the LTS give good the per capita GDP of Pakistan. It lowers the saving rate both estimates and are effected less as compared to LS estimates in at the domestic as well as national levels, whereas the sig- the presence of outliers. nificance of FDI reflects the truth that the economic de- velopment of Pakistan depends on the performance of FDI 6. Summary and Conclusion inflow up to some extent. Likewise, Table 3 shows that the LS analysis of the The current study has applied the least squares (LS) and the economic growth of India suggests that population growth high breakdown robust least trimmed squares (LTS) re- and gross savings are significant indicators for the economic gression techniques to estimate the impact of FDI inflow, growth of India. Population growth has a negative impact on annual population growth, and gross savings on the GDP per economic growth. One percentage increase of the annual capita of Pakistan and India. Within the LS framework, FDI population growth reduces the GDP per capita of India by exerts an insignificant and positive impact on the economic 2.03%, whereas the GDP per capita increases 0.14% with a growth of both Pakistan and India, whereas, after the ap- one percent increase in the gross savings. The impact of FDI plication of LTS technique, the FDI enters positively and inflow on the GDP per capita is positive but insignificant. significantly in the economic growth model of Pakistan, but The R2 and adj. R2 are very high and the regression seems to in the case of India, FDI has a negative insignificant impact be spurious, but as mentioned in Section 3, all the variables on the economic growth due to the elimination of 5 and 2 are cointegrated and a long-run equilibrium relationship outliers from the data of Pakistan and India, respectively. exists among variables and thus the regression model is not Population growth contributes to GDP per capita for both spurious. Thus, the overall regression model is highly sig- economies identically. Both techniques reveal that rapid nificant; 99.4% of the variation in GDP per capita is population growth negatively influences the economic explained by its linear relationship with the predictor var- growth of both countries and hence is a serious problem for iables. The significance of the regression model is also in- the economic growth of both economies, and it requires dicated by the residual standard error that takes a minimum immediate attention. Gross savings have a positive and value. insignificant impact on the economic growth of Pakistan, Robust LTS regression technique and plots of stan- whereas, for India, gross savings is the significant deter- dardized residuals and regression diagnostic reveal two minant of GDP per capita. Thus, sound economic policies outlying influential observations in the data with the stan- that improve and encourage the FDI inflow and gross dardized residuals of 3.4 and 3.9 in the absolute value savings in Pakistan are required to be formulated and corresponding to 1976 and 1979. Elimination of these two implemented. outliers improves the result. The gross savings is significant The application of LTS technique reveals that the overall at 5% level of significance in the classical model that is fit of the model improves, and the significance of the pre- significant at 1% with the robust regression. The impact of dictors changes significantly in both cases of Pakistan and FDI inflow though insignificant and positive in the classical India due to the removal of outliers from the data. Thus, model has turned out to be negative with LTS procedure, empirical results suggest that, to avoid the impact of bad data 8 Mathematical Problems in Engineering 3 2.5 Standardized LTS residuals 2 1 0 –1 –2 –2.5 –3 2 5 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 Fitted values Figure 5: Standardized residuals vs fitted values. Regression diagnostic plot 3 2.5 Standardized LTS residuals 2 1 41 32 36 0 37 40 33 31 34 35 6 38 39 –1 30 –2 –2.5 –3 2 5 1 2 3 4 5 6 Robust distance computed by MCD Figure 6: Standardized residuals vs robust distances. points and to avoid misleading results, the robust technique FinanceRetrived from https://papers.ssrn.com/sol3/papers. is strongly recommended. Results obtained through the cfm?abstract_id 1089964, December 2008. application of robust regression will largely help the A. A. Alamgir, S. A. Khan, D. M. Khan, and U. 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