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- 1. 11/23/2015 ANKITA MANDAL 001400302024 SAYANTAN BAIDYA 001400302042 SOUMI BHATTACHARYYA 001400302043 DEEPANWITA SAHA 001400302045 KRISHNENDU HALDER 001400302055 JADAVPUR UNIVERSITY DEPARTMENT OF ECONOMICS PG II SEMESTER III Simultaneous Equation System
- 2. Simultaneous Equation System November 23, 2015 1 | P a g e CONTENTS ACKNOWLEDGEMENT 2 ABSTRACT 3 INTRODUCTION 4 ECONOMIC THEORY 5 MODEL JUSTIFICATION 7 DATA USED 8 ANALYSIS 9 CONCLUSION 16 REFERENCE 17
- 3. Simultaneous Equation System November 23, 2015 2 | P a g e ACKNOWLEDGEMENT We are grateful to the faculty of Department of Economics (Jadavpur University) for their unwavering support and cooperation. Working on this project has given us the opportunity to gather immense knowledge regarding econometric tools and economic analysis that will surely benefit us significantly in our careers in the future. We thank our professor Dr. Arpita Dhar immensely for setting us this task of preparing and presenting this project. We are extremely grateful and thankful to her for her tireless guidance without which it would not have been possible for us to make progress in our endeavour. We also take this opportunity to thank our department for providing us with a functioning computer laboratory and library facilities which helped us to fulfil all our needs regarding our project. Moreover, we are also grateful to our friends and families for their constant support and help.
- 4. Simultaneous Equation System November 23, 2015 3 | P a g e ABSTRACT A casual look at the published empirical work in business and economics will reveal that many economic relationships are of the single-equation type.In such models, one variable (the dependent variable Y) is expressed as a linear function of one or more other variables (the explanatory variables, the X’s). In such models an implicit assumption is that the cause-and-effect relationship, if any, between Y and the X’s is unidirectional: The explanatory variables are the cause and the dependent variable is the effect. However, there are situations where there is a two-way flow of influence among economic variables; that is, one economic variable affects another economic variable(s) and is, in turn, affected by it (them). Thus, we need to consider twoequations. And this leads usto consider simultaneous-equation models, models in which there is morethan one regression equation, one for each interdependent variable. Keywords: Nonseparablemodels,simultaneousequations, multicollinearity. JEL classification. : C3.
- 5. Simultaneous Equation System November 23, 2015 4 | P a g e INTRODUCTION Single equation models, i.e., models in which there was a single dependent variable Y and one or more explanatory variables, the X’s, the emphasis was on estimating and/or predicting the average value of Y conditional upon the fixed values of the X variables. The cause-and-effect relationship, if any, in such models therefore ran from the X’s to the Y. But in many situations, such a one-way or unidirectional cause-and-effect relationship is not meaningful. This occurs if Y is determined by the X’s, and some of the X’s are, in turn, determined by Y. In short, there is a twoway, or simultaneous, relationship between Y and (some of) the X’s, which makes the distinction between dependent and explanatory variables of dubious value. It is better to lump together a set of variables that can be determined simultaneously by the remaining set of variables— precisely what is done in simultaneous-equation models. In such models there is more than one equation—one for each of the mutually, or jointly, dependent or endogenous variables. And unlike the single-equation models, in the simultaneous- equation models one may not estimate the parameters of a single equation without taking into account information provided by other equations in the system. In quite a similar view as of our single equation models , our conventional Classical Linear Model Assumption , is that all the explanatory variables of an equation are strictly unrelated. But when we start dealing with Simultaneity Bias it very likely gives rise to the problem of related explanatory variables, precisely the problem of Multicollinearity. In this particular problem the individual regression parameters are not estimable with sufficient precision because of high standard errors which often occurs due to highly intercorrelated regressors. In econometric literature Multicollinearity is one of the misunderstoood conceptions since high intercorrelations among the explanatory variables are neither necessary nor sufficient to cause the multicollinearity problem rather the best indicators of the problem are the t-ratios of the individual coefficients. Thus our endeavour is to resolve the problem of simultaneity, if present , as well as to emphasise on the fact that intercorrelation between explanatory variables is neither necessary nor sufficient for the existence of Multicollinearity rather we should concentrate more on the model’s significance.
- 6. Simultaneous Equation System November 23, 2015 5 | P a g e ECONOMIC THEORY OF SIMULTANEOUS EQUATION SYSTEM An obvious reason for the endogeneity of explanatory variables in a regression model is simultaneity: that is, one or more of the explanatory variables are jointly determined with the dependent variable. Models of this sort are known as simultaneous equations models (SEMs), and they are widely utilized in both applied microeconomics and macroeconomics. Each equation in a SEM should be a behavioral equation which describes how one or more economic agents will react to shocks or shifts in the exogenous explanatory variables, ceteris paribus. The simultaneously determined variables often have an equilibriuminterpretation, and we consider that these variables are only observed when the underlying model is in equilibrium. For instance, a demand curve relating the quantity demanded to the price of a good, as well as income, the prices of substitute commodities, etc. conceptually would express that quantity for a range of prices. But the only price-quantity pair that we observe is that resulting from market clearing, where the quantities supplied and demanded were matched, and an equilibrium price was struck. In the context of labor supply, we might relate aggregate hours to the average wage and additional explanatory factors: where the unit of observation might be the county. This is a structural equation, or behavioral equation, relating labor supply to its causal factors: that is, it reacts the structure of the supply side of the labor market. This equation resembles many that we have considered earlier, and we might wonder why there would be any difficulty in estimating it. But if the data relate to an aggregate such as the hours worked at the county level, in response to the average wage in the county this equation poses problems that would not arise if, for instance, the unit of observation was the individual, derived from a survey. Although we can assume that the individual is a price- (or wage-) taker, we cannot assume that the average level of wages is exogenous to the labor market in Suffolk County. Rather, we must consider that it is determined within the market, affected by broader economic conditions. We might consider that the z variable expresses wage levels in other areas, which would cet.par. have an effect on the supply of labor in Suffolk County; higher wages in Middlesex County would lead to a reduction in labor supply in the Suffolk County labor market, cet. par. To complete the model, we must add a specification of labor demand: where we model the quantity demanded of labor as a function of the average wage and additional factors that might shift the demand curve. Since the demand for labor is a
- 7. Simultaneous Equation System November 23, 2015 6 | P a g e derived demand, dependent on the cost of other factors of production, we might include some measure of factor cost (e.g. the cost of capital) as this equation's z variable. In this case, we would expect that a higher cost of capital would trigger substitution of labor for capital at every level of the wage, so that Note that the supply equation represents the behavior of workers in the aggregate, while the demand equation represents the behavior of employers in the aggregate. In equilibrium, we would equate these two equations, and expect that at some level of equilibrium labor utilization and average wage that the labor market is equilibrated. These two equations then constitute a simultaneous equations model (SEM) of the labor market. Neither of these equations may be consistently estimated via OLS, since the wage variable in each equation is correlated with the respective error term. How do we know this? Because these two equations can be solved and rewritten as two reduced form equations in the endogenous variables hi and wi. Each of those variables will depend on the exogenous variables in the entire system-z1 and z2-as well as the structural errors ui and vi: In general, any shock to either labor demand or supply will affect both the equilibrium quantity and price (wage). Even if we rewrote one of these equations to place the wage variable on the left hand side, this problem would persist: both endogenous variables in the system are jointly determined by the exogenous variables and structural shocks. Another implication of this structure is that we must have separate explanatory factors in the two equations. If z1 = z2; for instance, we would not be able to solve this system and uniquely identify its structural parameters. There must be factors that are uniqueto each structural equation that, for instance, shift the supply curve without shifting the demand curve. The implication here is that even if we only care about one of these structural equations-for instance, we are tasked with modelling labor supply, and have no interest in working with the demand side of the market-we must be able to specify the other structural equations of the model. We need not estimatethem, but we must be able to determine what measures they would contain. For instance, consider estimating the relationship between murder rate, number of police, and wealth for a number of cities. We might expect that both of those factors would reduce the murder rate, cet.par.: more police are available to apprehend murderers, and perhaps prevent murders, while we might expect that lower-income cities might have greater unrest and crime. But can we reasonably assume that the number of police (per capita) is exogenous to the murder rate? Probably not, in the sense that cities striving to reduce crime will spend more on police. Thus we might consider a second structural equation that expressed the number of police per capita as a function of a number of factors. We may have no interest in estimating this equation (which is behavioral, re effecting the behavior of city officials), but if we areto consistently estimate the former equation-the behavioral equation reeffecting the behaviorof murderers{we will have to specify the second equation as well, and collect data for its explanatory factors.
- 8. Simultaneous Equation System November 23, 2015 7 | P a g e MODEL AND ITS ECONOMIC JUSTIFICATION We have to deal with either seemingly unrelated regression equation or Simultaneous Equation System where we have to search for the problem of related explanatory variables and the factors which might give rise to the problem of multicollinearity. So we have taken into consideration a Simultaneous Equation System :- Money demand :- Md=β0 + β1Yt + β2Rt + β3 Pt+ u1t Money supply :-Ms = α0 + α1Yt + u2t Where M= Money Y= Income / GDP (gdp) R=Treasury Bill Rate (tbrate) P=Consumer price index(cpi) This model has a justifiable economic implication. The first equation shows that Money Demand is the dependent variable and it depends on GDP , Interest rate or treasury bill rate and consumer price index. The economy’s gross domestic product obviously defines the money demand since people will be demanding money based on the income generated in the economy. Money demand depends on interest rate since if the interest rate is lower people will be demanding more money and if the interest rate is higher people will be demanding less money. Theconsumer price index states the consumption basket of the consumers and hence it depends on price and income of the consumers and on how much money is available with them thus in return making money demand dependent on consumer price index. The second equation shows that Money Supply depends on the economy’s income only because depending on the income generated the financial authority decides whether to restrict money supply or not.
- 9. Simultaneous Equation System November 23, 2015 8 | P a g e MONEY, GDP, INTEREST RATE, AND CONSUMER PRICE INDEX, INDIA, 1970–1999 OBSERVATION M2 GDP TBRATE CPI 1970 626.4 3578 6.458 38.8 1971 710.1 3697.7 4.348 40.5 1972 802.1 3998.4 4.071 41.8 1973 855.2 4123.4 7.041 44.4 1974 901.9 4099 7.886 49.3 1975 1015.9 4084.4 5.838 53.8 1976 1151.7 4311.7 4.989 56.9 1977 1269.9 4511.8 5.265 60.6 1978 1365.5 4760.6 7.221 65.2 1979 1473.1 4912.1 10.041 72.6 1980 1599.1 4900.9 11.506 82.4 1981 1754.6 5021 14.029 90.9 1982 1909.5 4913.3 10.686 96.5 1983 2126 5132.3 8.63 99.6 1984 2309.7 5505.2 9.58 103.9 1985 2495.4 5717.1 7.48 107.6 1986 2732.1 5912.4 5.98 109.6 1987 2831.1 6113.3 5.82 113.6 1988 2994.3 6368.4 6.69 118.3 1989 3158.4 6591.9 8.1 124 1990 3277.6 6707.9 7.51 130.7 1991 3376.8 6676.4 5.42 136.2 1992 3430.7 6880 3.45 140.3 1993 3484.4 7062.6 3.02 144.5 1994 3499 7347.7 4.29 148.2 1995 3641.9 7543.8 5.51 142.4 1996 3813.3 7813.2 5.02 156.9 1997 4028.9 8159.5 5.07 160.5 1998 4380.6 8515.7 4.81 163 1999 4643.7 8875.8 4.66 166.6 Notes: M2 = M2 money demand (billions of dollars, seasonally adjusted). GDP = gross domestic product (billions of dollars, seasonally adjusted). TBRATE = 3-month treasury bill rate, %. CPI = Consumer Price Index (1982–1984 = 100). Source: Economic Report of the President, 2001, Tables B-2, B-60, B-73, B-69.
- 10. Simultaneous Equation System November 23, 2015 9 | P a g e ANALYSIS Since we have simultaneous equation model so it is highly likely that we will have a problem of identification and the model can be just identified,over-identified or under-identified. For dealing with problems of identification we use methods of Indirect least squares for just identified models and method of two stage and three stage least square for over identified and under identified models. In our simultaneous equation model we have M2(Money demand) and GDP(gdp) as the endogenous variables and treasury bill rate (tbrate) and consumer price index (cpi) as the exogenous variables. Since GDP is itself a dependent variable thus it gives rise to identification problem of simultaneity and from the model it is evident that here we have a problem of overidentification. Overidentification can be resolved by computing two stage least square regression of the entire simultaneous equation model for which it is required to obtain the reduced form of GDP in terms of the two exogenous variables tbrate and cpi. . ge ey = 2662.307 + 34.62386*cpi - 59.71936 t r e _cons 2662.307 237.2587 11.22 0.000 2175.493 3149.122 cpi 34.62386 1.375115 25.18 0.000 31.80236 37.44537 tbrtae -59.71936 22.66009 -2.64 0.014 -106.214 -13.2247 gdp Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 67039652.9 29 2311712.17 Root MSE = 299.52 Adj R-squared = 0.9612 Residual 2422168.66 27 89709.9504 R-squared = 0.9639 Model 64617484.2 2 32308742.1 Prob > F = 0.0000 F( 2, 27) = 360.15 Source SS df MS Number of obs = 30 . reg gdp tbrtae cpi
- 11. Simultaneous Equation System November 23, 2015 10 | P a g e Now we are generating gdp10 which will be dependent on only the two exogenous variables and hence if we replace gdp by gdp10 then the simultaneity problem will be resolved since the gdp present in the equation of money demand will no longer be correlated with the error term and neither the other explanatory variables gen gdp10 = 2662.307 + 34.62386*cpi - 59.71936*tbrate gdp10 will be dealt as an instrument of gdp and thus for computing 2SLS regression we regress m2 on tbrate cpi and the replaced value of gdp which is gdp10. As for example if Yt is an endogenous variable it can be expressed by two variables – one is the estimated Yt or Yt^ which is free of the error term and the estimated error Vt^. Yt = Yt^+ Vt^ The instrumental variable regression / 2SLS regression Instruments: tbrtae cpi gdp10 Instrumented: gdp _cons -1839.792 359.1825 -5.12 0.000 -2576.774 -1102.81 cpi 7.665082 5.819632 1.32 0.199 -4.275816 19.60598 tbrtae 0 (omitted) gdp .5948185 .1629644 3.65 0.001 .2604433 .9291938 m2 Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 42797787 29 1475785.76 Root MSE = 128.64 Adj R-squared = 0.9888 Residual 446782.742 27 16547.509 R-squared = 0.9896 Model 42351004.2 2 21175502.1 Prob > F = 0.0000 F( 2, 27) = 1276.91 Source SS df MS Number of obs = 30 Instrumental variables (2SLS) regression . ivreg m2 (gdp=gdp10) tbrtae cpi
- 12. Simultaneous Equation System November 23, 2015 11 | P a g e While computing the instrumental variable regression or the two stage least square regression we take gdp10 as the instrumented gdp. Since we have data available only for money demand so we run the two stage least square regression on the equation of money demand replacing gdp by the instrument gdp10. Here we see that we get the desired signs of all the coefficients and also the variables are quite significant.. So the 2SLS regression gives us desired results and is successful in abolishing the problem of over identification. Now we come to the Three Stage Least Square Regression After conducting the three stage least square regression we see that all the coefficients have desired signs and all the variables are significant at 1% level of significance and only tbrate is significant at 5% level of significance. The adjusted R2 value is also high showing that the goodness of fit is highly significant. Exogenous variables: gdp tbrate cpi Endogenous variables: m2 _cons -1132.728 180.7951 -6.27 0.000 -1487.08 -778.3761 cpi 16.8606 2.17858 7.74 0.000 12.59066 21.13054 tbrate -15.86045 8.135697 -1.95 0.051 -31.80612 .0852237 gdp .3292353 .0616228 5.34 0.000 .2084567 .4500138 m2 m2 Coef. Std. Err. z P>|z| [95% Conf. Interval] m2 30 3 95.90556 0.9936 4623.01 0.0000 Equation Obs Parms RMSE "R-sq" chi2 P Three-stage least-squares regression . reg3 ( m2 gdp tbrate cpi)
- 13. Simultaneous Equation System November 23, 2015 12 | P a g e MULTIC0LLINEARITY AND ITS INDICATOR The term multicollinearity is due to Ragnar Frisch. Originally it meant the existence of a “perfect,” or exact, linear relationship among some or all explanatory variables of a regression model. For the k-variable regression involving explanatory variable X1, X2, . . . , Xk(where X1 = 1 for all observationsto allow for the intercept term), an exact linear relationship is said toexist if the following condition is satisfied: λ1X1 + λ2X2 +· · ·+λkXk= 0 whereλ1, λ2, . . . , λkare constants such that not all of them are zero simultaneously. Today, however, the term multicollinearity is used in a broader sense to include the case of perfect multicollinearity as well as the case where the X variables are intercorrelated but not perfectly so, as follows : λ1X1 + λ2X2 +· · ·+λ2Xk + vi= 0 wherevi is a stochastic error term. If multicollinearity is perfect , the regression coefficients of the X variables are indeterminate and their standard errors are infinite. If multicollinearity is less than perfect, the regression coefficients, although determinate, possess large standard errors (in relation to the coefficients themselves), which means the coefficients cannot be estimated with great precision or accuracy. There are several sources of multicollinearity :- 1. The data collection method employed, for example, sampling over a limited range of the values taken by the regressors in the population. 2. Constraints on the model or in the population being sampled. For example, in the regression of electricity consumption on income (X2) and house size (X3) there is a physical constraint in the population in that families with higher incomes generally have larger homes than families with lower incomes. 3. Model specification, for example, adding polynomial terms to a regression model, especially when the range of the X variable is small. 4. An overdetermined model. This happens when the model has more explanatory variables than the number of observations. This could happen in medical research where there may be a small number of patients about whom information is collected on a large number of variables.
- 14. Simultaneous Equation System November 23, 2015 13 | P a g e For our simultaneous equation model its quite likely to have a multicollinearity problem since explanatory variable of one of the equations is the dependent variable of the other and that is why it’s quite probable that the explanatory variable has a relation with the other explanatory variables and error term. In our example gdp is the explanatory cum dependent variable which gives rise to both the problem of simultaneity and multicollinearity. As is evident from the regression table below we see that gdp has correlation with tbrate and cpi and all the coefficients are noticeably significant. So our equation of money demand will obviously have problem of multicollinearity. Here we are providing individual regression of Money Demand with each of its explanatory variable and we see that for all the 3 cases the coefficients give desired values at significant confidence intervals. We have conducted this test in order to see whether when we run a combined regression the values get hampered or not because of multicollinearity. _cons -2195.468 126.646 -17.34 0.000 -2454.89 -1936.045 gdp .7911096 .0211633 37.38 0.000 .7477586 .8344606 m2 Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 42797787 29 1475785.76 Root MSE = 173.28 Adj R-squared = 0.9797 Residual 840727.688 28 30025.9889 R-squared = 0.9804 Model 41957059.3 1 41957059.3 Prob > F = 0.0000 F( 1, 28) = 1397.36 Source SS df MS Number of obs = 30 . reg m2 gdp . ge ey = 2662.307 + 34.62386*cpi - 59.71936 t r e _cons 2662.307 237.2587 11.22 0.000 2175.493 3149.122 cpi 34.62386 1.375115 25.18 0.000 31.80236 37.44537 tbrtae -59.71936 22.66009 -2.64 0.014 -106.214 -13.2247 gdp Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 67039652.9 29 2311712.17 Root MSE = 299.52 Adj R-squared = 0.9612 Residual 2422168.66 27 89709.9504 R-squared = 0.9639 Model 64617484.2 2 32308742.1 Prob > F = 0.0000 F( 2, 27) = 360.15 Source SS df MS Number of obs = 30 . reg gdp tbrtae cpi
- 15. Simultaneous Equation System November 23, 2015 14 | P a g e _cons 3411.099 611.5584 5.58 0.000 2158.379 4663.82 tbrtae -153.0497 85.76094 -1.78 0.085 -328.7231 22.62359 m2 Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 42797787 29 1475785.76 Root MSE = 1171.5 Adj R-squared = 0.0701 Residual 38426955.7 28 1372391.27 R-squared = 0.1021 Model 4370831.3 1 4370831.3 Prob > F = 0.0852 F( 1, 28) = 3.18 Source SS df MS Number of obs = 30 . reg m2 tbrtae _cons -548.9967 80.42128 -6.83 0.000 -713.7322 -384.2612 cpi 28.80403 .7313971 39.38 0.000 27.30583 30.30223 m2 Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 42797787 29 1475785.76 Root MSE = 164.64 Adj R-squared = 0.9816 Residual 758942.463 28 27105.088 R-squared = 0.9823 Model 42038844.5 1 42038844.5 Prob > F = 0.0000 F( 1, 28) = 1550.96 Source SS df MS Number of obs = 30 . reg m2 cpi _cons -1132.728 194.2051 -5.83 0.000 -1531.922 -733.5337 cpi 16.8606 2.340171 7.20 0.000 12.05031 21.67089 tbrtae -15.86045 8.73914 -1.81 0.081 -33.82401 2.103111 gdp .3292353 .0661935 4.97 0.000 .1931725 .465298 m2 Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 42797787 29 1475785.76 Root MSE = 103.02 Adj R-squared = 0.9928 Residual 275936.319 26 10612.9353 R-squared = 0.9936 Model 42521850.6 3 14173950.2 Prob > F = 0.0000 F( 3, 26) = 1335.54 Source SS df MS Number of obs = 30 . reg m2 gdp tbrtae cpi
- 16. Simultaneous Equation System November 23, 2015 15 | P a g e But we see even when we compute a combined regression the factors remain significantly different from zero with coefficients having desired values and signs. Though this is an elementary way of checking whether multicollinearity poses a big problem or not but we can somehow observe that multicllinearity is not a big adversity in this case. But a more conventional and methodical approach is to find out the extent and intensity of the multicollinearity problem for which we have to conduct the Variance Inflation Factor(VIF) test where VIF = 1/1-R2i Where R2i gives the goodness of fit or the degree of association of the ith variable with all the other explanatory variables and thus higher the value of R2i higher is the degree of association between the explanatory variables which states multicollinearity. So if VIF value is high that means the extent of the problem is high and if the value is low then the extent of the problem is low. Here we see that none of the VIF values are high which says that the degree of multicollinearity is not much and also we see that while regressing we get the desired signs of the coefficients and also the variables are significant with mostly 1% level of significance and for tbrate we have 10% level of significance. So we need not consider the problem of multicollinearity even if it is present in the model but has not much adverse effect. Herein once again our model proves that though VIF indicates high inter- correlation among the explanatory variables still our t-ratios are significant showing that related regressosr is not an efficient clause determining multi-collinearity. The model becomes acceptable even after the presence of the problem of multi-collinearity. Mean VIF 18.39 tbrtae 1.34 0.744731 cpi 26.15 0.038247 gdp 27.68 0.036130 Variable VIF 1/VIF . estat vif
- 17. Simultaneous Equation System November 23, 2015 16 | P a g e CONCLUSION Thus in our simultaneous equation model we see that as per our pre conceived idea there arises a problem of identification. We have a problem of over identification and we resolve it through instrumental variable regression where we take the help of two stage and three stage least square regression to find out the desired estimates. As per our model we find that our regression estimates are significant in each case and the R2s are high showing that the regression result fits the model properly. In the same model we have to check for the presence of multicollinearity. Since our model comprises of money demand , money supply , GDP , interest rate and consumer price index so its very well evident that if we deal with the equation of money demand where GDP , interest rate and consumer price index are the explanatory variables , we are likely to get a relation between the explanatory variables and specially when GDP is the explanatory variable for the equation of money supply too. GDP(variable name gdp) is the variable which gives rise to both the problem of simultaneity and multicolllinearity. Thus elementarily we conduct the individual regression of the dependent variable money demand with each of its independent variable and find that all the regressors are significant and when we conduct the combined regression then also we find that the regressors remain significant and the level of significance also remains the same. So from simple regression we can conclude that multicollinearity is not posing a big problem though we are sure that it is present. Thus we carry on the more conventional VIF test where we see that VIF estimates are quite high rather more than 5 except for tbrate. From our usual knowledge of econometrics we would say that there is high collinearity between the estimates but in reality we see that multi collinearity is not creating a problem for the regression which is running smoothly with proper and significant estimates. .
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