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An econometric model for Linear Regression using Statistics
1.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 886 An econometric model for Linear Regression using Statistics Renvil Dsa1, Remston Dsa2 1Renvil Dsa Fr. Conceicao Rodrigues College of Engineering, Mumbai, India 2Remston Dsa New York University, New York, USA ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - This research paper discusses the econometric modeling approach of linear regression using statistics. Linear regression is a widely used statistical technique for modeling the relationship between a dependent variable and one or more independent variables. The paper begins by introducing the concept of linear regression and its basic assumptions. The univariate and multivariate linear regression models are discussed, and the coefficients of the regression models are derived using statistics. The matrix form of the Simple Linear Regression Model is presented, and the properties of the Ordinary Least Squares (OLS) estimators are proven. Hypothesis testing for multiple linear regression is also discussed in the matrix form. The paper concludes by emphasizing the importance of understanding the econometric modeling approach of linear regression using statistics. Linear regression is a powerful tool for predicting the values of the dependent variable based on the values of the independent variables, and it can be applied in various fields, including economics, finance, and social sciences. The paper's findings contribute to the understanding of the linear regression model's practical application and highlight the need for rigorous statistical analysis to ensure the model's validity and reliability. Key Words: econometric model, linear regression, statistics, univariate regression, multivariate regression, OLS estimators, matrix form, hypothesis testing. 1. INTRODUCTION Linear regression is a powerful statistical modeling technique widely used to analyze the relationship between a dependent variable and one or more independent variables. In a simple linear regression model, the dependent variable is assumed to be a linear function of one independent variable, while in a multiple linear regression model, the dependent variable is a linear function of two or more independent variables. The coefficients of the linear regression model are represented by beta (β) values, which are constants that determine the slope and intercept of the regression line. The beta values are estimated using the Ordinary Least Squares (OLS) method, which involves minimizing the sum of squared errors between the predicted values and the actual values of the dependent variable. The matrix form of the linear regression model is a useful tool for understanding the relationship between the dependent and independent variables. The matrix form allows for a more efficient calculation of the beta values and the variance-covariance matrix of the beta values. Partial derivatives are used to derive the expected values and variation of the beta values. The expected values of the beta values are equal to the true beta values, and the variation of the beta values can be used to derive confidence intervals and hypothesis tests for the beta values. The relationship between the beta values and the normal distribution is important in understanding the properties of the OLS estimators. The beta values are normally distributed, and their variance-covariance matrix can be used to derive confidence intervals and hypothesis tests. Hypothesis testing is an essential component of linear regression analysis. The null hypothesis is typically that the beta value is equal to zero, indicating that there is no relationship between the independent variable and the dependent variable. The t-test and F-test are commonly used to test hypotheses about the beta values. The ANOVA matrix is a useful tool for decomposing the total variation in the dependent variable into the explained variation (sum of squared regression) and unexplained variation (sum of squared error). The sum of squared total is the sum of squared deviations of the dependent variable from its mean. The relationship between the sum of squared regression, sum of squared error, and sum of squared total with the mean squared error and the degrees of freedom is important in interpreting the results of the linear regression analysis. In summary, linear regression is a powerful statistical modeling technique that can be used to analyze the
2.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 887 relationship between a dependent variable and one or more independent variables. The beta values, matrix form of the model, partial derivative concept, expected values and variation of beta values, relationship with normal distribution, hypothesis testing, ANOVA matrix, formulas for sum of squared regression, sum of squared error, sum of squared total, and relationship with mean squared error, t-test, and F-test with degrees of freedom are important concepts in understanding and interpreting the results of linear regression analysis. 2. METHODS 2.1 Deriving the coefficient of the regression model for the simple linear regression model The important result that is needed to derive ̂ and ̂ Suppose, is the sample mean of y then is defined as 1 over n times the summation from i = 1 to n of = ∑ It also means that is equal to the summation from i = 1 to n of = ∑ ------------- ( a ) Similarly, is the sample mean of y and is defined as 1 over n times the summation from i = 1 to n of = ∑ It also means that is equal to the summation from i = 1 to n of . = ∑ ------------ ( b ) There are some important results that we need to know using the summation operator, ∑ ( - )( - ) = The written L.H.S. can also be written as: ∑ - -------- (1) ∑ ( - ) -------- (2) ∑ ( - ) -------- (3) Table 1: Different forms of ∑ ( - )( - ) [1] Derive: ∑ = ∑ - L.H.S. = ∑ = ∑ ( - - + ) = ∑ - ∑ - ∑ + From equation (a) and equation (b) = ∑ - + - = ∑ - = R.H.S [Equation (1)] Derive: ∑ = ∑ ( - ) L.H.S. = ∑ = ∑ ( - - + ) = ∑ - ∑ - ∑ + From equation (b) ∑ - - ∑ + ∑ - ∑ ∑ ( - ) = R.H.S [Equation (2)]
3.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 888 Derive: ∑ = ∑ ( - ) L.H.S. = ∑ = ∑ ( - - + ) = ∑ - ∑ - ∑ + From equation (a) ∑ - ∑ - + ∑ - ∑ ∑ ( - ) = R.H.S. [Equation (3)] Deriving the formula for ̂ and ̂ [2] 1. S.S.E: The sum of squared error is equal to the summation from i = 1 to n of the square of the difference between and ̂ 2. is the real observation and ̂ is the predicted observation 3. The goal is to minimize the distance between the real observation and the predicted observation. 4. The equation is a quadratic equation and the objective is to find the minimum point and not the maximum point. S.S.E = ∑ ( ̂ ) substituting ̂ = ̂ ̂ S.S.E = ∑ ( ̂ ̂ ) Applying Partial Derivative with respect to ̂ ̂ = 2 ∑ ( ̂ ̂ ) (-1) To find the maxima or minima, Equivalently, the derivative equals 0. 0 = - 2∑ ( ̂ ̂ ) 0 = ∑ - ̂ - ̂ ∑ from equation (a) and equation (b) 0 = - ̂ - ̂ 0 = - ̂ - ̂ ̂ = - ̂ Applying Partial Derivative with respect to ̂ = - 2 ∑ ( ̂ ̂ ) To find the maxima or minima, Equivalently, the derivative equals 0. = - 2 ∑ ( ̂ ̂ ) 0 = ∑ - ̂ ∑ - ̂ ∑ From equation (b) 0 = ∑ - ̂ - ̂ ∑ As ̂ = - ̂ 0 = ∑ - ̂ - ̂ ∑ 0 = ∑ - + ̂ - ̂ ∑ From equation (1) 0 = ∑ + ̂ - ∑ ) - ̂ - ∑ ) = ∑ ̂ = ∑ ∑ As ∑ = (∑ ) ̂ = ∑ ∑ 2.2 Matrix form of the Simple Linear Regression Model
4.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 889 Here, is a 2 x 2 symmetric matrix det( ∑ ∑ ∑ = n∑ - [∑ = n∑ - = n(∑ - n ) = ( ∑ - n ) = ∑ det( ) = n∑ = n ̂ ̂ X Scalar form of Y is = + For Y, the intercept, is the slope, and is the error term epsilon. This is known as the scalar form. Suppose there are n data points, then i ranges from 1 to n The Matrix Form of Y is [3] 2.3 SLR Matrix Form Proof of Properties of OLS(Ordinary Least Squares) Estimators The first assumption is that the expected value of epsilon is equal to zero for all i and these i are in the range of i = 1, 2,..., n, where n is the number of data points. The mathematical representation is stated below. E[ ] = 0 i, where i = 1, 2, 3,..., n. The anticipated value of vector epsilon is going to be zero vector or in rows, so it's essentially epsilon, the expected value of epsilon is going to be simply a vector of zero, where there will be n rows, So this is the first assumption that will be evaluated when working with the simple linear regression model in matrix form. Expected value and Variance of [4] E[ ] = The second assumption is about the variance of the epsilon Var[] i. The epsilon vector's variance matrix is equal to sigma squared times the identity matrix, with dimensions of n by n.
5.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 890 Var[ ] = So the variance of epsilon is equal to sigma squared on all diagonal terms up to n rows, and all these entries should be zero other than diagonal terms. Var[ ] = I The third assumption is that the epsilon values are uncorrelated. If they are uncorrelated then that means that for any Epsilon values and , the variance of and is equal to zero. This also means that the covariances of and are equal to zero. Thus, where i j, the covariance of and is equal to zero. As ’s are uncorrelated The mathematical representation of the above third assumption is: Var( ) = 0 Cov( ) = 0 Cov( ) = 0 i As the epsilon of i's( ’s) are uncorrelated with each other, and based on these three assumptions, when combined with the fourth assumption, allow for a very good powerful construction of the simple linear regression model, from which various results can be derived. The fourth and last assumption is that the error terms are normally distributed. The epsilon i( ) follows the normal distribution with some mean( ) and some variance( ) . N( , ) [5] From Assumptions 1 and 2, we know that = E[ ] = and Var[ ] = where is the standard deviation. Also, from assumption 3 where i, where i = 1, 2, 3,........., n Cov( ) = 0 i which means these epsilon are independent and are identically distributed. Hence, N( , ) Hence, ̂ = Y - ̂ ̂ = Y - ̂ Derive and show that E[ ̂] = As ̂ = E[ ̂] = E[ ] Note: is not stochastic, it’s not random at all, it is all just constant known values that are multiplied by each other. E[ ̂] = E[ ] As, Y = X + E[Y] = E[X + ] = E[X ] + E[ ] = E[X ] ----(From Assumption 1, E[ ] = 0) = X Hence, E[ ̂] = E[ ] = ( X) As ( X) = where X is symmetric matrix E[ ̂] = E[ ̂] = Derive and show that Var[ ] = ̂ Var[ ̂] = Var[ ] Let B = Var[B ] = B Var[ ] = B Var[X + ] = B Var[ ] + B Var[ ] Here, in Var[ ] is not random and continuous hence we can assume that Var[ ] = 0
6.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 891 From Assumption 2 where Var[ ] = I Var[ ̂] = B I = I As is symmetric, = X = X Var[ ̂] = I = I X = As ( X) = where X is symmetric matrix = I = As are unobservable, cannot be computed hence we assume ̂is an estimator of Hence, Var[ ̂] = ̂ 2.4 Hypothesis testing for Multiple Linear Regression- Matrix form Y = X here, ̂ = Y - ̂ where Y above in the equation is the actual observation and ̂ is the fitted observation. The following equation gives the sum of squared errors(SSE): SSE = ̂ ̂ Therefore, the above function can be written as: SSE = ̂ ̂ To fit the regression model, the goal should be to minimize the SSE, minimize(SSE) = minimize(̂ ̂) = minimize[ ̂ ̂ ] Here, ̂ = where E[ ̂] = Var[ ̂]= ̂ ̂ ̂ = ̂ = MSE = = ̂ ̂ Here, p regressors of interest 1 is for and n is the number of samples in the population Let Se be the Standard Error Hence, Se( ̂) = √ ̂ ̂ = √ ̂ Here, Se( ̂ ) = √ ̂
7.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 892 Hypothesis to be tested: : ̂ = 0 vs : ̂ 0 [6] Test Statistic = ̂ ̂ ̂ = ̂ ̂ = ̂ √̂ Hypothesis testing on a linear combination of regression parameters in the matrix form of the linear regression problem. Below is the general formula of the regression model that is in matrix form: Y = X Scalar representation of the regression parameters: Y = + + + + ……….+ . Linear combination: Let ̂ = + + +...............+ . The , , ,...., in the above equation represent the constants. ̂ ̂ Hypothesis to be tested: :̂ = vs : ̂ Expected value of ̂ : E[̂ ] = E[̂ ̂] = E[ ̂] = = L is the row vector of the constants. Variance of ̂ : ̂ = = c = ̂ c Formulation of the test statistics is a t-test where the parameter of interest (̂) follows the normal distribution with mean of L and variance of ̂ c. ̂ N( , ̂ c ) Test statistics T: T = ̂ ̂ : It is the value of the linear combination under the null hypothesis where ̂ = √ ̂ Then test statistics becomes, T = √̂ ANOVA Matrix Form [7]: SSR : Sum of squares Regression : ∑ (̂ ) SSE : Sum of squares Errors : ∑ ̂ SST : Sum of Squared Total : ∑ ( ) Source of variation Sum of Squares degree s of freedo m Mean Square F statistic Regresion SSR p MSE = F = Errors SSE n-p-1 MSE = = ̂
8.
International Research Journal
of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 10 Issue: 04 | Apr 2023 www.irjet.net p-ISSN: 2395-0072 © 2023, IRJET | Impact Factor value: 8.226 | ISO 9001:2008 Certified Journal | Page 893 Total SST = SSR + SSE n-1 MST = Table 2: ANOVA matrix form [8] F = Hypothesis to be tested: : = = = =...= vs : F [9] f F > then reject else accept 3. CONCLUSIONS In conclusion, this research paper has explored the econometric modeling approach of linear regression using statistics. The paper has presented the fundamental concepts of linear regression, such as the basic assumptions and the definition of the dependent and independent variables. The paper has then discussed the univariate and multivariate linear regression models, and the coefficients of the regression models have been derived using statistical methods. The matrix form of the Simple Linear Regression Model has been presented, and the properties of the Ordinary Least Squares (OLS) estimators have been proven. Finally, the paper has discussed hypothesis testing for multiple linear regression in the matrix form. The importance of understanding the econometric modeling approach of linear regression using statistics has been emphasized, and the paper's findings contribute to the understanding of the practical application of the linear regression model. Overall, linear regression is a powerful tool for predicting the values of the dependent variable based on the values of the independent variables. It can be applied in various fields, including economics, finance, and social sciences. Rigorous statistical analysis is necessary to ensure the model's validity and reliability, and this paper has provided insights into the statistical methods used in linear regression modeling. REFERENCES [1] Introduction to Linear Regression Analysis, 5th Edition" by Montgomery, Peck, and Vining [2] Applied Linear Regression, 3rd Edition" by Weisberg [3] Matrix Approach to Simple Linear Regression" by Kato and Hayakawa [4] Linear Regression Analysis: Theory and Computing" by Liu and Wu [5] Linear Models with R, 2nd Edition" by Faraway [6] Applied Regression Analysis and Generalized Linear Models, 3rd Edition" by Fox [7] Statistical Models: Theory and Practice" by Freedman, Pisani, and Purves [8] Statistical Inference" by Casella and Berger [9] Probability and Statistics for Engineering and the Sciences, 9th Edition" by Devore
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