1. ELEMENTS OF HIERARCHICAL REGRESSION LINEAR MODELS
CHAPTER 22
ELEMENTS OF HIERARCHICAL REGRESSION LINEAR MODELS
Damodar Gujarati
Econometrics by Example, second edition

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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