Intuition for Random Effects

Previously I wrote a post based on course notes from J.Blumenstock that attempted to provide some intuition for how fixed effects estimators can account for unobserved heterogeneity (individual specific effects).

Recently someone asked if I could provide a similarly motivating and intuitive example regarding random effects. Although I was not able to come up with a new example, I can definitely discuss random effects in the same context of the previous example. But first a little (less intuitive) background.

Background

To recap, the purpose of both fixed and random effects estimators is to model treatment effects in the face of unobserved individual specific effects.

y_it =b x_it + α_i + u_it  (1) 

In the model above this is represented by α_{i .} In terms of estimation, the difference between fixed and random effects depends on how we choose to model this term. In the context of fixed effects it can be captured through a dummy variable estimation (this creates different intercepts or shifts capturing specific effects) or by transforming the data, subtracting group (fixed effects) means from individual observations within each group.  In random effects models, individual specific effects are captured by a composite error term (α_i + u_it) which assumes that individual intercepts are drawn from a random distribution of possible intercepts. The random component of the error term α_i captures the individual specific effects in a different way from fixed effects models. 

As noted in another post, Fixed, Mixed, and Random Effects, the random effects model is estimated using Generalized Least Squares (GLS) :

β_GLS = (X’Ω^-1X)^-1(X’Ω^-1Y) where Ω = I ⊗ Σ    (2) 

Where Σ is the variance α_i+ u_it . If  Σ is unknown, it is estimated, producing a feasible generalized least squares estimate β_FGLS

Intuition for Random Effects

In my post Intuition for Fixed Effects I noted: 

"Essentially using a dummy variable in a regression for each city (or group, or type to generalize beyond this example) holds constant or 'fixes' the effects across cities that we can't directly measure or observe. Controlling for these differences removes the 'cross-sectional' variation related to unobserved heterogeneity (like tastes, preferences, other unobserved individual specific effects). The remaining variation, or 'within' variation can then be used to 'identify' the causal relationships we are interested in."

Lets look at the toy data I used in that example. 

The crude ellipses in the plots above (motivated by the example given in Kennedy, 2008) indicate the data for each city and the the 'within' variation exploited by fixed effects models (that allowed us to correctly identify the correct price/quantity relationships expected in the previous post). The differences between the ellipses represents 'between variation.' As Kennedy discusses, random effects models differ from fixed effects models in that they are able to exploit both 'within' and 'between' variation, producing an estimate that is a weighted average of both kinds of variation (via Σ in equation 2 above). OLS, on the other hand exploits both kinds of variation as an unweighted average.

More Details 

As Kennedy discusses, both FE and RE can be viewed as running OLS on different transformations of the data.

For fixed effects: "this transformation consists of subtracting from each observation the average of the values within its ellipse"

For random effects: "the EGLS (or FGLS above) calculation is done by finding a transformation of the data that creates a spherical variance-covariance matrix and then performing OLS on the transformed data."

As Kennedy notes, the increased information used by RE makes them more efficient estimators, but correlation between 'x' and the error term creates bias. i.e. RE assumes that α_i is uncorrelated with (orthogonal to) regressors. Angrist and Pischke (2009) discuss (footnote, p. 223) that they prefer FE because the gains in efficiency are likely to be modest while the finite sample properties of RE may be worse. As noted on p.243 an important assumption for identification in FE is that the most important sources of variation are time invariant (because information from time varying regressors gets differenced out). Angrist and Pischke also have a nice discussion on page 244-245 discussing the choice between FE and lagged dependent variable models.

References:

A Guide to Econometrics. Peter Kennedy. 6th Edition. 2008
Mostly Harmless Econometrics. Angrist and Pischke. 2009

See also: ‘Metrics Monday: Fixed Effects, Random Effects, and (Lack of) External Validity (Marc Bellemare.

Marc notes: 

"Nowadays, in the wake of the Credibility Revolution, what we teach students is: “You should use RE when your variable of interest is orthogonal to the error term; if there is any doubt and you think your variable of interest is not orthogonal to the error term, use FE.” And since the variable can be argued to be orthogonal pretty much only in cases where it is randomly assigned in the context of an experiment, experimental work is pretty much the only time the RE estimator should be used."

Intuition for Fixed Effects

I've written about fixed effects before in the context of mixed models. But how are FE useful in the context of causal inference? What can we learn from a panel data using FE that we can't get from a standard regression with cross sectional data?  Let's view this through a sort of parable, based largely on a very good set of notes produced by J. Blumenstock, used in a management statistics course (link).

Suppose we have a restaurant chain and have gathered some cross sectional data on the pricing and consumption of large pizzas for some portion of the day for some period 1 across three cities, as pictured below:

Now, if we are trying to infer the relationship between price and quantity demanded using this data, we notice something odd. The theoretically implied negative relationship does not exist. In fact, if we plot the points, this seems more in line with a supply curve rather than a demand curve:

What's going on that could explain this? One explanation could be specific individual differences across cities related to taste and quality. Perhaps in Chicago, customer's tastes and preferences are for much more expensive and higher quality pizza, and they really like pizza a lot. They may be willing to pay more for more pizzas aligned with their specific tastes and preferences. Perhaps this is also true for San Francisco, but to a lesser extent, and in Atlanta maybe not so much.

What we have is unobserved heterogeneity related to these specific individual effects. How can we account for this? Suppose we instead collected the same data for two periods, essentially creating a panel of data for pizza consumption:

Now, if we look 'within' each city, the data reveals the theoretically implied relationship between price and demand. Take San Francisco for example:

This is essentially what fixed effects estimators using panel data can do. They allow us to exploit the 'within' variation to 'identify' causal relationships. Essentially using a dummy variable in a regression for each city (or group, or type to generalize beyond this example) holds constant or 'fixes' the effects across cities that we can't directly measure or observe. Controlling for these differences removes the 'cross-sectional' variation related to unobserved heterogeneity (like tastes, preferences, other unobserved individual specific effects). The remaining variation, or 'within' variation can then be used to 'identify' the causal relationships we are interested in.

See also: Difference-in-Difference models. These are a special case of fixed effects also used in causal inference.

Reference:
Fixed Effects Models(Very Important Stuff)
www.jblumenstock.com/courses/econ174/FEModels.pdf

GMM, Endogeneity, SNA, Viral Marketing, and Causal Inference

 In the article “Impact of social network structure on content propagation: A study using YouTube data” the authors investigate the relationship between socioemetric measures like degree centrality with diffusion of videos across a network.  In other words, they wanted to know if there was a causal relationship between network properties of those that share videos and the likelihood that a video would become viral.  What first interested me about this article was that it was a very good example of an application of social network analysis and viral seeding.  However, it also provides some very good examples of applications related to generalized method of moments,  instrumental variables, unobserved heterogeneity and endogeneity, and causal inference.  I previously was not aware of the GMM style of dynamic panel data models that instrument with lags, which is apparently quite popular in many econometric applications (see references below).

As the authors point out, any model that relates network properties to the outcome of video dissemination requires a careful estimation strategy if we are interested in making causal inferences. They identity several sources of endogeneity and unobserved heterogeneity.  If we are trying to infer dissemination based on one’s position in the network, we have to consider that other unobserved factors related to network position and video type could also impact dissemination.  It may be the case that all we are trying to do is predict video shares based on network position,  and perhaps that is OK as long as these correlations hold over time.

In contrast, if we want to make causal inferences, these types of endogeneity must be accounted for and also make econometric estimation difficult. In this case what we really want to estimate is the independent causal effect of network position on video shares, so we are interested only in the ‘quasi-experimental’ variation in network position.

A natural solution involves an instrumental variables approach, but the challenge of finding an ‘external’ instrument that is correlated with network and video properties of interest, but uncorrelated with unobserved effects is rather daunting. Ultimately the authors propose a generalized method of moments dynamic panel estimator using lagged variables as instruments.  

References:

 Anderson, T. W., & Hsaio, C. (1981). Estimation of dynamic models with error components. Journal of the American Statistical Association, 76(375), 598–606.

Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Economic Studies, 58, 277–97.

DYNAMIC PANEL DATA MODELS:

A GUIDE TO MICRO DATA METHODS AND PRACTICE

Stephen Bond

THE INSTITUTE FOR FISCAL STUDIES

DEPARTMENT OF ECONOMICS, UCL

cemmap working paper CWP09/02

Impact of social network structure on content

propagation: A study using YouTube data

Quant Mark Econ (2012) 10:111–150

Hema Yoganarasimhan

Mixed, Fixed, and Random Effects Models

The General Mixed Model:

Y  = Xβ + Z α+ e            (1)

β =  fixed effects co-efficient vector

X = fixed effects model matrix

α = random effects co-efficient matrix

Z =  random effects model matrix

The fixed part of the model is specified by Xβ and the random part by Zα+ e.

 The ‘random effects’ matrix (α) represents random effects that vary across individuals vs. the ‘fixed effects’ matrix (β) that represents effects that are the same across all individuals.

Panel Data: Cross sectional time series data, in most cases looking at hundreds or thousands of individuals (units) observed at several points across time, i.e. multiple observations per unit across time.

Heterogeneity Effect: Whether or not effects, or responses of individuals are the same across time, or if there are group differences.  If effects are not the same, and they are not accounted for, estimation errors result. Fixed and random effects models attempt to capture the heterogeneity effect. 

Given

y_it =b x_it + α_i + u_it                  (4)

 α_i is an unobserved individual effect

Fixed Effects Model (FE): α_i is correlated with x. In this model individual effects, or differences across individuals can be captured by shifts in the regression equation, or dummy variables.

As such we can estimate the fixed effects model as a Least Squares Dummy Variable model (LSDV):

y= Xb + dα + e            (5)

where d is a vector of dummy variables for each individual or unit effect.

Random Effects Model (RE): α_i is uncorrelated with x. Individual effects are randomly distributed across units.

y_it =b x_it + α_i+ u_it            (6)

Note that (4) and (6) are identical, the only difference is the assumptions about the individual effect α_i and how the models are estimated.

Note also that (6) differs from (1) in that we don’t have the matrix Z. The Random Effects model is in fact a special case of the general mixed model with a random intercept (α_i).

The random effects model is estimated using Generalized Least Squares (GLS) :

β_GLS = (X’Ω^-1X)^-1(X’Ω^-1Y) where Ω = I ⊗ Σ    (7)

Where Σ is the variance α_i+ u_it . If  Σ is unknown, it is estimated, producing a feasible generalized least squares estimate β_FGLS

Whether a FE or RE model should be used can be determined based on the Housman test (see Greene, for more details)

References:

Greene, Econometric Analysis, 5^th Edition