Introduction to Fixed Effects Methods
Costs and Benefits of Fixed Effects Methods
Fixed effects methods in nonexperimental research are highly valued for their ability to control for stable individual characteristics, effectively reducing bias Within-subject comparisons, commonly used in changeover or crossover designs, involve subjects receiving various treatments at different times, with response variables measured for each treatment Randomizing the order of treatments is ideal, as the primary goal of crossover designs is to minimize sampling variability, thereby enhancing the power of hypothesis testing.
Differencing out individual variability among subjects helps to reduce error variance commonly found in traditional experimental designs, where each subject is exposed to a single treatment.
When fixed effects methods are used on nonexperimental data, they often lead to increased sampling variability compared to other analytical approaches This occurs because, in typical observational studies, independent variables can fluctuate both within individual subjects and across different subjects For instance, if personal income is measured annually over five years, significant variation may be observed within a single individual's income over time, but most of the variation is expected to occur between different individuals.
Fixed effects methods concentrate solely on within-person variation, disregarding between-person variation This approach can lead to significantly higher standard errors compared to methods that consider both variations However, focusing on within-person variation helps eliminate contamination from unmeasured personal characteristics correlated with income, resulting in more accurate and unbiased estimates.
In analyzing nonexperimental data, there is a trade-off between bias and sampling variability, where fixed effects methods often reduce bias but increase sampling variability This trade-off is generally favorable due to the expected bias in observational studies However, a significant limitation of fixed effects methods occurs when the within- to between-person variance ratio approaches zero, rendering it impossible to estimate coefficients for variables lacking within-subject variation, such as race, sex, or region of birth Additionally, these methods are less effective in estimating the impact of stable characteristics like height or years of schooling among adults Despite this, fixed effects regression effectively controls for these stable variables, even without direct measurements, and allows for the inclusion of interactions between stable and time-varying variables Ultimately, fixed effects methods are best suited for examining the effects of variables that exhibit within-subject variation in most observational studies.
In experimental data, bias is minimized by applying the same treatments to all subjects and randomizing their order, resulting in nearly zero correlation between treatments and stable subject characteristics, eliminating the need for fixed effects Additionally, since all variations in independent variables occur within subjects, focusing on this within-subject variation retains all necessary information Consequently, utilizing fixed effects methods can significantly reduce standard errors due to a smaller variance in the error term.
Why Are These Methods Called “Fixed Effects”?
The name “fixed effects” is a source of considerable confusion As we shall see, the basic idea is very simple Consider the linear model ij i ij ij x
In the linear model represented by Y = β0 + β1 + α + ε, the subscripts i and j indicate different individuals and measurements over time, respectively The term β1 xij is considered a fixed effect, as it involves measured values with β1 as a constant parameter Conversely, εij is treated as a random variable, assumed to follow a normal distribution with a mean of 0 and a variance of σ² Thus, this typical linear model incorporates both fixed and random components, highlighting the complexity of statistical analysis in various contexts.
The term α i represents the stable characteristics of individuals, and there is a critical decision to be made regarding its treatment as either fixed or random Mixed models, such as those estimated by PROC MIXED, consider α i as a random variable with a specified probability distribution, typically normal and independent of measured variables, known as random effects models in econometrics Conversely, fixed effects models treat α i as a set of fixed parameters, which can be directly estimated or conditioned out during the estimation process, hence the term "fixed effects."
The choice between fixed effects and random effects methods depends on your research objectives While fixed effects methods have their own set of advantages and disadvantages, random effects methods present a contrasting set Unlike fixed effects, random effects do not control for unmeasured, stable characteristics of individuals, as they assume that these characteristics are uncorrelated with the measured variables However, random effects allow for the estimation of stable covariates like race and gender, and they typically exhibit less sampling variability due to their use of variation within and between individuals Although this book primarily focuses on fixed effects methods, it frequently contrasts them with random effects approaches.
In my view, then, the decision to treat the between-person variation as fixed or random should depend largely on
• whether it’s important to control for unmeasured characteristics of individuals
• whether it’s important to estimate the effects of stable covariates
• whether one can tolerate the substantial loss of information that comes from discarding the between-individual variation
In the literature on ANOVA and experimental design, however, the decision between fixed and random effects is often described in quite different terms Consider the following characterization:
In experimental research, treatment effects are typically considered fixed when inferences are limited to the specific treatment levels tested However, if the analysis aims to generalize beyond the tested levels or if the treatment levels were not deliberately chosen, it is standard to treat the treatment effects as random (LaMotte 1983, pp 138–139).
In this context, treating the between-person variation, represented by α i, as random may seem beneficial for making broader population inferences However, this perspective is flawed for two main reasons First, the inclusion of α i in the equation primarily aims to estimate the coefficients of other variables while controlling for unmeasured covariates and addressing the lack of independence in multiple observations per person, making broader generalizations irrelevant Second, fixed effects models are typically less restrictive than random effects models, allowing for a more accurate representation of the data Notably, in linear models, random effects estimators are essentially a specific instance of fixed effects estimators.
Fixed Effects Methods in SAS/STAT
Different types of dependent variables necessitate specific fixed effects methods, and SAS offers various procedures for implementing these methods For linear models, PROC GLM is the most user-friendly for fixed effects analysis, while PROC REG and PROC MIXED can be utilized with additional effort In fixed effects logistic regression, PROC LOGISTIC is the preferred choice, although PROC GENMOD can be applied when there are only two observations per individual For fixed effects Poisson regression, PROC GENMOD is the sole option available When it comes to fixed effects survival analysis, PROC PHREG is the recommended procedure, though PROC LOGISTIC may also be applicable in certain situations Lastly, PROC CALIS is beneficial for fitting fixed effects linear models that include lagged endogenous variables.
What You Need to Know
To effectively engage with this book, your prior knowledge will depend on your intended depth of understanding For instance, to grasp chapter 2 on linear models, familiarity with multiple linear regression is essential, including an understanding of its assumptions and the estimation process through ordinary least squares Ideally, you should possess considerable practical experience with multiple regression applied to real data and be comfortable interpreting regression analysis outputs Additionally, a solid grasp of fundamental statistical inference concepts—such as standard errors, confidence intervals, hypothesis testing, p-values, bias, and efficiency—is crucial.
To read chapter 3, you should have, in addition to the requirements for chapter 2, a knowledge of logistic regression at about the level of the first three chapters of my 1999 book
Logistic regression is a key statistical method used for binary outcomes, and understanding its fundamental model is crucial for effective analysis This article explores the theory behind binary logistic regression, detailing the process of estimating the model using maximum likelihood estimation Additionally, it emphasizes the importance of interpreting results through odds ratios While prior knowledge of PROC LOGISTIC can enhance comprehension, it is not a prerequisite for grasping the concepts presented.
For chapter 4 on fixed effects Poisson regression, you should have a basic familiarity with the Poisson regression model, discussed in chapter 9 of Logistic Regression Using the SAS
System: Theory and Application I’ll use PROC GENMOD to estimate the model, so previous experience with this procedure will be helpful
Chapter 5 on survival analysis emphasizes the importance of understanding the Cox proportional hazards model and partial likelihood estimation These fundamental concepts are thoroughly explained in my 1995 book, "Survival Analysis Using SAS: A Practical Guide," which also provides detailed instructions on utilizing the PHREG procedure.
To effectively understand chapter 6, it is essential to have a foundational knowledge of linear structural equation models (SEMs) that encompass both observed and latent variables For SAS users seeking a comprehensive introduction to this subject, "A Step-by-Step Approach to Using the SAS System for Factor Analysis and Structural Equation Modeling" by Hatcher (1994) is highly recommended.
This book aims to minimize the use of mathematics, avoiding calculus and limiting matrix notation However, to enhance the clarity of regression models, vector notation is utilized frequently.
Having some knowledge of maximum likelihood estimation can be beneficial, but it is not essential for understanding the content Familiarity with SAS/STAT and the SAS DATA step will enhance your ability to follow the presented SAS programs However, the majority of the programs in this book are straightforward and concise, making them accessible even for beginners in SAS.
Computing
This book presents all computer input and output generated by SAS 9.1 for Windows, highlighting differences in syntax from earlier versions SAS programs are formatted with keywords in upper case, while user-defined variable and data set names are in lower case In the main text, both SAS keywords and user-specified variables are consistently shown in upper case.
The displays in this article were generated using the SAS Output Delivery System To maintain focus, the ODS statements are not included in the program examples provided However, readers interested in replicating these displays can use the specified code before and after the procedure statements.
OPTIONS LSu PS000 NODATE NOCENTER NONUMBER;
ODS RTF FILE='c:\book.rtf' STYLE=JOURNAL BODYTITLE;
All the examples were run on a Dell Optiplex GX270 desktop computer running Windows
XP at 3 gigahertz with 1 gibabyte of physical memory
Fixed Effects Methods for Linear Regression
2.2 Estimation with Two Observations Per Person 10
2.4 Estimation with PROC GLM for More than Two Observations Per Person 19
2.5 Fixed Effects versus Random Effects 25
2.7 An Example with Unbalanced Data 38
In Chapter 1, I demonstrated that the traditional paired-comparisons t-test can be viewed as a fixed effects approach, accounting for all stable individual characteristics This model serves as a specific instance of a broader linear model designed for quantitative response variables, which is the focus of this chapter.
In this article, we denote the response variable for individual i at time t as y For clarity, we will refer to individuals as persons and the occasions as different times of measurement It's important to note that in certain contexts, i may represent groups while t could indicate various individuals within those groups.
In our analysis, we incorporate predictor variables where \( z_i \) represents a column vector of time-invariant characteristics describing individuals, while \( x_{it} \) denotes a column vector of variables that fluctuate over time.
The foundational model employed in this analysis is represented by the equation y_it = α_t + β x_it + γ z_i + α_i + ε_it, where i ranges from 1 to n and t spans from 1 to T This model accounts for individual-specific effects and time variations, allowing for a comprehensive understanding of the relationships between the variables involved For those unfamiliar with vectors, these can also be interpreted as single variables.
In this equation, \( t \) represents a time-varying intercept, while \( \beta \) and \( \gamma \) are row vectors of coefficients The term \( \epsilon \) denotes a random disturbance, and \( \alpha_i \) accounts for all stable individual differences not captured by \( \gamma z_i \).
In a fixed effects model, we regard these as fixed parameters, one per person That implies that x it may be correlated with α i
In this article, we maintain the conventional linear model assumptions, including a mean of zero for ε it, constant variance, and no covariance between ε it and ε jt for i ≠ j Unlike traditional linear regression, we do not require ε it to be uncorrelated with z i or α i However, we impose stronger assumptions regarding the relationship between ε it and x it, specifically that x it is strictly exogenous This implies that x it at any time t is statistically independent of random disturbances across all time points, meaning it cannot be influenced by y at any earlier times In chapter 6, we will relax this assumption to explore potential reciprocal relationships between the two variables.
Under the given assumptions, equation (2.1) can be effectively estimated using ordinary least squares (OLS) The implementation of OLS for the fixed effects model varies based on the data structure and requires addressing several special considerations To illustrate this, we will first examine the simpler scenario where each individual has precisely two measurements for both the response variable and the time-varying predictor variables.
2.2 Estimation with Two Observations Per Person
When analyzing data with two observations per individual (t = 1, 2), the fixed effects model can be efficiently estimated through OLS regression by utilizing difference scores for all time-varying variables The relevant equations for these two time points are essential for this analysis.
Subtracting the first equation from the second, we get
In this analysis, both γz i and α i have been eliminated from the equation, preventing the estimation of γ, the coefficients for time-invariant predictors However, this approach effectively controls for the influence of these variables Additionally, if the error terms ε i1 and ε i2 meet the standard linear model assumptions, their difference will also adhere to these assumptions, even if they are correlated Thus, applying Ordinary Least Squares (OLS) to the difference scores will yield unbiased and efficient estimates of β, the coefficients for time-varying predictors.
In this analysis, we examine data from a sample of 581 children who participated in interviews conducted in 1990, 1992, and 1994 as part of the National Longitudinal Survey of Youth (Center for Human Resource Research, 2002) Our focus will be on three specific variables that were consistently measured across all three interview years.
ANTI antisocial behavior, measured with a scale ranging from 0 to 6
SELF self-esteem, measured with a scale ranging from 6 to 24
POV poverty status of family, coded 1 for in poverty, otherwise 0
In this section, we will use only the data from 1990 and 1994 Our goal is to estimate a regression model of the form:
The model presented, ANTI(t) = à(t) + β1SELF(t) + β2POV(t) + ε(t) for t = 1, 2, posits that both poverty and self-esteem at time t simultaneously influence antisocial behavior While acknowledging potential uncertainties regarding the causal relationships among these variables, this chapter will set aside those complexities Additionally, due to the limited data points, estimating a model that incorporates both lagged variables and fixed effects is not feasible It is also assumed that the regression coefficients remain constant across time points, although this assumption can be tested or modified as discussed later.
To estimate the regression equation for each year, I utilized PROC REG on the SAS data set MY.NLSY, which contains one observation per person and separate variables for measurements across different years The following SAS code was employed to estimate the model.
PROC REG DATA=my.nlsy;
Please note that this and all other data sets used in this book are available for download at support.sas.com/companionsites
In the analysis presented, both independent variables demonstrate statistical significance beyond the 05 level for the years studied Notably, higher self-esteem correlates with reduced antisocial behavior, while poverty is linked to increased antisocial behavior Furthermore, the impact of self-esteem (SELF) was more pronounced in 1994 compared to 1990, whereas the influence of poverty (POV) showed a decline over the same period.
Output 2.1 Regressions of ANTI on POV and SELF in 1990 and 1994
Dependent Variable: anti90 child antisocial behavior in 1990
Dependent Variable: anti94 child antisocial behavior in 1994