Fixed effects model

Fixed effects models are a type of regression model used to analyze the effect of one or more independent variables on a dependent variable while controlling for the effects of other variables. They assume that the effects of certain variables on the dependent variable remain constant over time and across different groups or individuals. Fixed effects models are useful for simplifying a complex analysis and for reducing the risk of bias caused by omitted variables. They are often used in fields such as economics, finance, and marketing to explain the effects of certain policies or factors on the outcomes of interest.

Example of fixed effects model

• A fixed effects model could be used to understand the effect of a price change on the demand for a product. By controlling for other factors, such as the product's quality, advertising, or economic conditions, the researcher can isolate the effect of the price change and determine whether it had a positive or negative effect on demand.
• In the field of finance, a fixed effects model could be used to identify the factors that influence the returns on a portfolio of stocks. By controlling for market returns, the researcher can isolate the effects of individual stocks on the portfolio's performance.
• In the field of economics, a fixed effects model could be used to investigate the effect of tax policy changes on economic growth. By controlling for other factors, such as government spending, the researcher can isolate the effect of the tax policy change and determine whether it had a positive or negative effect on economic growth.

Formula of fixed effects model

The general formula of a fixed effects model is:

$$Y_{it} = \beta_0 + \beta_1X_{1it} + ... + \beta_kX_{kit} + \alpha_i + \varepsilon_{it}$$

where:

$$Y_{it}$$ is the value of the dependent variable observed at time t for individual i,

$$X_{1it}, ..., X_{kit}$$ are the values of the k independent variables observed at time t for individual i,

$$\beta_0$$ is the intercept,

$$\beta_1, \beta_2, ..., \beta_k$$ are the coefficients of the k independent variables,

$$\alpha_i$$ is the fixed effect for individual i,

$$\varepsilon_{it}$$ is an error term.

The fixed effect for individual i, $$\alpha_i$$, captures the effect of all the omitted variables that are specific to individual i, such as personal characteristics or unobservable factors. This term is assumed to be constant over time and across different individuals.

The coefficients of the independent variables, $$\beta_1, \beta_2, ..., \beta_k$$, capture the effects of these variables on the dependent variable, controlling for all other variables, including the individual-specific fixed effects. These parameters are estimated using methods such as maximum likelihood estimation or ordinary least squares.

When to use fixed effects model

Fixed effects models are useful when attempting to explain the effects of certain policies or factors on the dependent variable while controlling for the effects of other variables. They can be used in fields such as economics, finance, marketing, and public policy. Specifically, fixed effects models can be used to:

• Analyze the effect of one or more independent variables on a dependent variable.
• Simplify a complex analysis and reduce the risk of bias caused by omitted variables.
• Explain the effect of changes in policies or factors on outcomes of interest.
• Estimate and compare the effects of different policies or factors on the dependent variable.
• Estimate the effects of interactions between different policies or factors.
• Estimate the effects of unobserved individual or group characteristics on the dependent variable.

Types of fixed effects model

Fixed effects models are a type of regression model used to analyze the effect of one or more independent variables on a dependent variable while controlling for the effects of other variables. Examples of fixed effects models include:

• Fixed-effects linear regression: this model is used to explain the effects of one or more independent variables on a single dependent variable. It assumes that the effects of the independent variables are constant over time and across different groups.
• Fixed-effects logistic regression: this model is used to explain the effects of one or more independent variables on a binary outcome (dependent variable). It allows for the estimation of odds ratios, which can be used to assess the strength of a relationship between an independent and a dependent variable.
• Fixed-effects ANOVA: this model is used to compare the means of multiple independent variables on a single dependent variable. It is similar to the linear regression model, but it allows for the estimation of the effect of each independent variable on the dependent variable.
• Multi-level fixed-effects models: this model allows for the estimation of the effects of multiple independent variables on a single dependent variable while controlling for the effects of other variables. It is a more complex version of the linear or logistic regression models and can be used to analyze data with a hierarchical structure.

Steps of fixed effects model

A fixed effects model is a regression model used to analyze the effect of one or more independent variables on a dependent variable while controlling for the effects of other variables. The following steps are involved in performing a fixed effects model:

• Identify the independent variables that may be affecting the dependent variable.
• Specify the functional form of the model, such as linear or logistic.
• Construct the model by specifying the parameters and the distributions of the independent variables.
• Estimate the model by fitting the data to the specified parameters.
• Evaluate the model by testing for goodness of fit, examining the estimated parameters, and checking for any violations of assumptions.
• Interpret the results by examining the estimated parameters and the predicted values of the dependent variable.
• Use the model to make predictions or decisions.

Limitations of fixed effects model

Fixed effects models have several drawbacks that should be considered before using them. These include:

• Lack of flexibility: Fixed effects models assume that the effects of the independent variables remain constant over time and across different groups or individuals. This can limit the ability to detect changes in the effect of the independent variables that may occur over time or in different contexts.
• Difficulty in interpreting results: Due to the nature of fixed effects models, results can be difficult to interpret. This is because the results are based on the differences between groups or individuals, rather than the absolute effects of the independent variables on the dependent variable.
• Limited number of independent variables: Fixed effects models typically allow for a limited number of independent variables due to the nature of the analysis. This can limit the ability to detect the full effect of the independent variables on the outcomes of interest.

Fixed effects model — recommended articles

Linear regression analysis — Analysis of covariance — Random effects model — Hierarchical regression analysis — Logistic regression model — Confirmatory factor analysis — Maximum likelihood method — Homogeneity of variance — Standardized regression coefficients

References

• Borenstein, M., Hedges, L. V., Higgins, J. P., & Rothstein, H. R. (2010). A basic introduction to fixed‐effect and random‐effects models for meta‐analysis. Research synthesis methods, 1(2), 97-111.