ELEMENTS OF HIERARCHICAL REGRESSION LINEAR MODELS CHAPTER 22 ELEMENTS OF HIERARCHICAL REGRESSION LINEAR MODELS Damodar Gujarati Econometrics by Example, second edition
HIERARCHICAL LINEAR MODELS (HLMs) Other names for these models (or ones with similar features) include: Multilevel models (MLM) Mixed-effect models (MEM) Random-effects models (REM) Random coefficient regression models (RCRM) Growth curve models (GCM) Covariance components models (CCM) Damodar Gujarati Econometrics by Example, second edition
Data often have a hierarchical structure: BASIC IDEA OF HLM Data often have a hierarchical structure: Micro-level, or lower-level, data are often embedded in macro-level, or higher-level, data. The primary goal of HLM is to predict the value of a micro-level dependent variable (i.e., regressand) as a function of other micro-level predictors (or regressors) as well as some predictors at the macro level. Damodar Gujarati Econometrics by Example, second edition
BASIC IDEA OF HLM (CONT.) Analysis at the micro level is Level 1 analysis. Analysis at the macro level is Level 2 analysis. Damodar Gujarati Econometrics by Example, second edition
HLM ANALYSIS OF THE NAÏVE MODEL In HLM, we assume: Where Y is the outcome, i is the micro-level observation, and j is the macro-level observation. We further assume that the random intercept is distributed around its mean value with the error term vj: Damodar Gujarati Econometrics by Example, second edition
HLM ANALYSIS OF THE NAÏVE MODEL We obtain: Composite error term wij is the sum of macro-specific error term vj (Level 2 error term) and micro-specific error term uij (Level 1 error term). Assuming these errors are independently distributed, we obtain the following variance: Damodar Gujarati Econometrics by Example, second edition
INTRA-CLASS CORRELATION COEFFICIENT The ratio of the macro-specific variance to the total variance is called the intra-class correlation coefficient (ICC): Gives the proportion of the total variation in Y attributable to the macro level. Higher ICC means macro differences account for a larger proportion of the total variance. So we cannot neglect the influence of macro differences. Damodar Gujarati Econometrics by Example, second edition