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A comprehensive guide to linear regression models, covering key concepts, model estimation, and testing procedures. It includes a practical application using spss to analyze factors affecting customer loyalty to hoa loc mango, demonstrating the steps involved in defining variables, entering data, performing regression analysis, and interpreting results. The document also explores the importance of independent variables and provides management implications for enhancing customer loyalty.
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LINEAR REGRESSION MODEL Business Administration CONTENT 2
**1. Introduction
1. INTRODUCTION 3 Linear regression is a fundamental concept in econometrics and is widely used in many different fields Suppose a researcher wants to understand the relationship between sales revenue and advertising costs. If they have specific data, they can estimate this relationship using a linear regression model 4
7 Y : Dependent variable (quantitative variable) α : Intercept (constant) βi (i = 1, 2, …, k): Slope coefficient for each variable Xi : Independent variables (quantitative or dummy variables) u : Error term (represents the variation in Y that cannot be explained by the variables Xi) 8 𝑌^ = ෝ α + β 1 𝑋 1 +^ β 2 𝑋 2 +.^.^.^ +^ β 𝑖𝑋𝑖 +^ 𝑢ො Estimated equation: 𝑢 ො: Residuals Use the Ordinary Least Squares (OLS) method in econometrics to estimate α and β Problem: Estimate α, β, and determine the linear correlation of Xi with Y => Use SPSS
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3. STEPS FOR TESTING THE LINEAR REGRESSION MODEL According to Green W.H. (1991), in the case where the number of observations is greater than 100 5 tests should be conducted. (1) Testing the partial correlations of the regression coefficients (2) Testing the goodness of fit of the model (3) Testing for multicollinearity (4) Testing for autocorrelation (5) Testing for heteroskedasticity 10 3. 1. Testing the partial correlations of the regression coefficients Whether the independent variables are significantly correlated with the dependent variable (considering each independent variable separately). T-test: The significance level (Sig.) of each partial regression coefficient with a 95% confidence level (sig. <= 0.05). It is also possible to choose 90% or 99%.
13 3. 4. Testing for autocorrelation Step 1: Determine the Durbin-Watson statistic (d) Step 2: Based on the Durbin-Watson table (considering the number of observations (n), number of parameters (k- 1 ), and a 5% significance level), determine dL (lower d) and dU (upper d) When the residuals are correlated with each other, the OLS estimation results are no longer reliable. 14 Figure 1 : Diagram for determining autocorrelation Positive autocorrelation No conclusion No autocorrelation No conclusion Negative autocorrelation 0 dL dU 4 - dU 4 - dL If dU < d < (4-dU) Conclusion: There is no autocorrelation in the residuals of the linear regression model.
15 3.5. Testing for heteroskedasticity The phenomenon where the residual values have different distributions. The OLS estimates of the regression coefficients are inefficient.
4. ESTIMATING THE LINEAR REGRESSION MODEL IN SPSS 16 Application Model: Factors Affecting Customer Loyalty to Hoa Loc Mango (Cai Be District, Tien Giang Province)
19 Data: Survey data of 100 customers in 2013 File SPSS: P 1 - DATA-TRUNGTXCHL-LEC
Estimated equation: Y: Dependent variable Xi: Independent variables u: Residuals 20 Determine the sample size for a regression model According to Green W.H. ( 1991 ), Tabachnick & Fidell ( 2007 )
(^) p: number of independent variables The research model has 5 independent variables and 1 dependent variable
21 Step 1: Define the variables in SPSS Step 2: Enter the data into SPSS Step 3: Perform regression analysis ANALYZE > REGRESSION > LINEAR Dependent: Enter variable KNMUA (Y) Independent: Enter variables X 1 ,X 2 ,X 3 ,X 4 ,X 5 Estimating the linear regression model in spss 22 Select the checked options in the Linear Regression: Statistics table. Select Continue/Save. Select Standardized/Continue Select OK.
25 4. 2. Testing the goodness of fit of the model Model's goodness of fit Analysis of Variance (ANOVA) With a confidence level of 99 % (Sig.< 0 , 01 ) => Theoretical model fits the data. => Independent variables are linearly correlated with the dependent variable 26 Explanation level Adjusted R^2 = 0 , 596 (F-Test, sig.<= 0 , 05 ) => 59. 6 % of the variation in Y is explained by the 5 independent variables.
27 4. 3. Testing for multicollinearity Variance Inflation Factor (VIF) VIF>10: There is multicollinearity The table above shows VIF< Conclusion: There is no multicollinearity. 28 4. 4. Testing for autocorrelation Durbin-Watson (d): 1, With 100 observations, the number of parameters (k- 1 ) = 5 , and a significance level of 0. 01 ( 99 %), refer to the Durbin-Watson statistic table: dL= 1 , 441 và dU= 1 , 647. dU<d= 1 , 888 <( 4 - dU= 2 , 353 ) No autocorrelation
31 4.5. Testing for heteroskedasticity Auxiliary regression model a X X X X X v a X a X a X a X a X u a aX a X aX a X a X ( * * * * ) ( ) ( ) ( ) ( ) ( ) 11 1 2 3 4 5 2 10 5 2 9 4 2 8 3 2 7 2 2 6 1 0 1 1 2 2 3 3 4 4 5 5 2 32 Target Variable: USQUARE Numeric Expresion: PHANDUPHANDU* The same applies to the other variables **X 12 ,...,X 52 , (X 1 *X 2 *X 3 X 4 X 5 ) Step 1: Calculate variables in auxiliary regression model Transform/Compute Variables
33 a X X X X X v a X a X a X a X a X u a aX a X aX a X aX ( * * * * ) ( ) ( ) ( ) ( ) ( ) 11 1 2 3 4 5 2 10 5 2 9 4 2 8 3 2 7 2 2 6 1 0 1 1 2 2 3 3 4 4 5 5 2 Step 2: Determine the auxiliary regression model Return to the SPSS interface ANALYZE/REGRESSION/LINER Enter the independent variables and the dependent variable into the model 34 Results of the auxiliary regression model R^2 : 0,241 => nR^2 =1000,241=24, With (k-1)=df1= parameters for the auxiliary regression model, a significance level of 0.01 (99%) in the Chi-square distribution table The critical value of Chi-square is 24. 72 n*R^2 = 24 , 1 < The critical value of Chi- square => Residual variance is constant
37 4.6. Discuss regression results Unstandardized regression coefficient. BX 1 = 0 , 217 (+): Positive relationship When customers rate safety (XCHL) one point higher, the likelihood of repurchase will increase by 0. 217 points. 38 BX 2 = 0 , 414 (+): Positive relationship The likelihood of repeat purchases for customers who know about HoaLoc Mango is 0. 414 points higher than for customers who do not have this information BX 4 = 0 , 132 (+): Positive relationship When customers rate satisfaction with price one point higher, the likelihood of repurchase will increase by 0. 132 points BX 5 = 0 , 235 (+): Positive relationship When customers rate their satisfaction with the convenience of their purchase by an additional 1 point, the likelihood of repurchase will increase by 0. 235 points.
39 Standardized regression coefficients Norusis ( 1993 ): Because the magnitude of the unstandardized regression coefficients (B) depends on the units of measurement of the variables, the B coefficients can only be compared when all the independent variables are measured in the same unit. In cases where the independent variables have different units of measurement, use Beta coefficients. Beta coefficients indicate the importance of the variables 40 Determine the importance of independent variables in percentage terms Standard. Beta % Thứ tự ảnh hưởng (X1) An toàn của XCHL 0,199 18,51 3 (X2) Thông tin, truyền thông về thương hiệu XCHL 0,471 43,81 1 (X4) Giá của XCHL 0,128 11,91 4 (X5) Thuận tiện khi mua XCHL 0,277 25,77 2 1,075 100,
