Making Inferences

Causal Inference

• Knowledge progresses by finding causes of phenomena
• e.g., Human activity $\Rightarrow ? \Rightarrow$ global warming

Hypothesis Testing

Perfectly Controlled Experimental Design (PCED)

• A Perfectly Controlled Experimental Design (PCED) represents in an abstract fashion the best way to test causal relations (Guala, 2005)
• A PCED is built around comparison and controlled variation
• Comparison → groups that have been exposed to different conditions are juxtaposed
• Controlled variation → all factors that are not intentionally manipulated should be kept constant across groups (uniformity)

Controlled Variation

• How to obtain a controlled variation?
1. Matching
• Units that are identical are assigned to groups in different conditions
• Consider a response variable $Y$
• $Y_t(u)$ is the value of the response that would be observed if the unit were exposed to $t$
• $Y_t(u)-Y_c(u)$ is the causal effect of $t$ (relative to $c$) on $u$
• Causal inference refers to the appraisal of $Y_t(u)-Y_c(u)$
• Fundamental Problem of Causal Inference: it is impossible to observe the value of $Y_t(u)$ and $Y_c(u)$ on the same unit and, therefore, it is impossible to observe the effect of $t$ on $u$

Controlled Variation (ii)

• How to obtain a controlled variation?
• 2. Randomization
• The random assignment of units to groups in different conditions makes the groups identical in expected terms

The identification framework (Newey, 2007)

• $Y_{i1}$ is the observed outcome for subject $i$ who is exposed to the intervention ($D_i=1$)
• $Y_{i0}$ is the observed outcome for subject $i$ who is NOT exposed to the intervention ($D_i=0$)
• The outcome $Y$ for individual $i$ is thus $Y_i=Y_{i0}+(Y_{i1}-Y_{i0})D_i=\alpha_i+\beta_iD_i$, with $\alpha_i=Y_{i0}$ and $\beta_i=Y_{i1}-Y_{i0}$
• The average tretament effect (ATE) is the average of the slope ($\beta$) over the entire population
• This is generally what the experimenter is interested in
• The average tretament effect on treated (ATT) is the average of the slope ($\beta$) over the population of treated ($D_i=1$)

The identification framework (Newey, 2007)

• If
• the mean of $Y_{i1}$ does not depend on the treatment status ($E[Y_{i1}|D_i]=E[Y_{i1}]$)
• and
• the mean of $Y_{i0}$ does not depend on the treatment status ($E[Y_{i0}|D_i]=E[Y_{i0}]$)
• → $ATE=ATT=E[\beta_i]$
• This condition holds when individuals are randomly assigned to treatments
• Being assigned to a treatment does not depend on the output variable $Y_i$
• In field happenstance data, it is likely that $E[Y_{i}|D_i]\neq E[Y_{i}]$ (self selection into a treatment)
• Solved econometrically via instrumental variables or other "selection on observables" techniques.

Models of Experimental Design

• Basic models of experimental design (guala, 2005)
• Post-test

• The (causal) effect is given by $Y_{G_1}-Y_{G_2}$
• Pre-test/post-test

• The (causal) effect is given by $(O_{G_2}-O_{G_1})-(O_{G_4}-O_{G_3})$ (diff in diff)
• The Pre-test/post-test allows to control for the impact of measurement

Experimental Economics

Why Running Economic Experiments?

• Roth (1995) identifies 3 main motives for experiments in Economics
• Speaking to Theorists
• Testing predictions originating from formal theories
• The organization of empirical regularities into formal theories
• Searching for Facts (Meaning)
• The study of effects that are not (yet) part of a well-structured theory
• The accumulation of facts that may lead to the creation of new theories (``Searching for Meanings'')
• Whispering to the Ears of Princes
• The formulation of advices for policy makers

Methodological Issues

• Hertwig and Ortmann (2001) identify 4 essential methodological aspect of experiments in economics
• Script Enactment
• Comprehensive instructions reduce risk of ``demand effects''
• Favors replicability
• Repeated trials
• Allows for learning and, potentially, equilibrium convergence
• Performance-based monetary payments
• Proper monetary payments allow to control for the disturbance due to cognitive effort
• Deception
• Deception may induce suspicion among participants and generate uncontrolled reputation spillovers

Validity of Experimental Findings

• Methodological aspect affect the validity of the experiment
• The validity of an experiment can be assessed along two dimensions
• Internal validity
• The extent to which an experiment allows to draw inferences about the behavior in the experiment
• External validity
• The extent to which an experiment allows to draw inferences about behavior outside the experiment

Internal Validity

`Just as we need to use clean test tubes in chemistry experiments, so we need to get the laboratory conditions right when testing economic theory'' (Binmore, 1999)

• To ensure internal validity a proper experimental setting should be implemented
• Examples of dirty tubes
• Bad incentive schemes
• Individuals self-select into treatments
• Deceitful instructions
• Experimenter effects
• Reputation (supergames)

External Validity

• 3 dimensions of artificiality that may impact on external validity of experiments (Bardsley et al., 2010)
1. Artificiality of isolation
• The experimental setting may omit relevant features of the external environment
• Findings in the lab may not survive when transferred to the complexity of the real-world
2. Artificiality of omission and contamination
• The laboratory environment may bias results (e.g., bad incentives)
• A lack of internal validity produces a lack of external validity

External Validity (ii)

• 3. Artificiality of alteration
• The laboratory alters natural phenomena
• Individuals attach a different meaning to a market in the laboratory and to a real stock market $\rightarrow$ it is impossible to study stock markets in the laboratory
• Artificiality of alteration is the most challenging

Building Blocks

Building Blocks

• Repetition
• One shot vs. repeated trials
• Stationary repetition, conditional task, ...
• Payment scheme
• Random lottery incentive, cumulative payment ...
• Choice Task (stage game)
• Roles, action space, ...
• Matching
• Partner matching, absolute stranger, perfect stranger, ...

Repetitions

• One-shot: the stage game is repeated only $1 \times$

• Round	ID
  1	101	102	103	104

• Repeaterd trials: the stage game is repeated $N>1 \times$

• Round	ID
  1	101	102	103	104
  2	101	102	103	104
  3	101	102	103	104
  ...	101	102	103	104

• Repetitions can be
• Stationary: same task over distinct rounds
• Dynamic: different tasks over distinct rounds

Incentive-compatibility

• Motivation of participants is controlled via (monetary) rewards $\Rightarrow$ Induced Value Theory (Smith, 1976)
• 3 qualifications of the postulate
• Subjective costs (values) may be non-negligible and interfere with monetary reward
• e.g., cognitive effort may overcome the monetary incentives
• A game value may be attached to experimental (nominal) outcomes (e.g., points or tokens)
• As long as the value function is monotone it does not interfere with induced valuation (reinforcement)
• Individuals may be concerned with others' utility and thus not be independent in their evaluation
• Increasing one's own payoff not necessarily increases utility (e.g., equity concerns)

Payment schemes

• Different payment schemes could be adopted when repeated choices are collected
• Cumulative payment
• All choices are rewarded
• Random Lottery Incentive (RLI)
• One choice is randomly chosen for payment
• Chosen round is communicated to participants only at the end of the expeirment
• The RLI mechanism generates incentive compatible choices as long as participants are EUT
It avoids potential confounds due to wealth effects or edging wrt cumulative payment

Choice Task

• Aspects of the choice task that need to be carefully defined
• Roles in the interaction should be assigned to participants
• Usually roles are randomly defined by randomly assign participants to computer terminals
• Action space
• The action space affects the nature of the outcome variable
• Dichotomous data, Count data, Continuous data, String data
• Payoff rule
• Must produce incentive compatible choices
• Important to check that participants understood it
• When tokens are employed a conversion rate should be specified

Data Independence

• Statistical tests should be run on independent observations
• The outcome of one observational unit should not be affected by the outcome of another observational unit
• When choices of one participant interact with choices of another participant their choices are no more independent
• The contagion analogy
• This is true even when the interaction is indirect but mediated by a bridging individual

An Example

• 4 players interact in groups of 2 over two rounds
• Blue and Red
• Thus, the following connections are created

• ID	1	2	3	4
  1	1
  2	1	1
  3	1	0	1
  4	0	1	1	1

• 3 and 2 are not directly connected, but indirectly (via Pl. 4)
• Only one independent observation in the experiment!
• Average choice of the four players

Matching Protocol

• 16 Participants
• ID: 1, 2, ... 16
• 4 groups
• 
• Blue, Red, Orange, Green
• 4 rounds
• Round 1, Round 2, Round 3, Round 4
• Each group has 4 participants

Partner Matching

• 4 participants interact in the 4 rounds
• 
• Players 1-4 are always in the same (blue) group
• Reutation building

Random Partner Matching

• Each group is made of 4 participants and participants in a group interact together
• At each repetition the groups are formed randomly
• 
• e.g., player 1 is in group Blue in rounds 1--3 and in group Green in round 4
• Two players can be matched together in different rounds
• e.g., 1 and 9 are in the same group in rounds 1 and 4
• PRO: simple to implement and to explain; control on reputation concerns
• CONS: the number of independent observations is endogenous and shrinks with repetitions

Random Partner Matching with Matching Groups

• 4 groups: Blue, Red, Orange, Green
• The groups are matched into two subgroups
• [Blue, Red] and [Orange, Green]
• Each group is made of 4 participants and participants in a group interact together
• In each repetition the groups are formed randomly, within the subgroups
• 
• Participants in the two subgroups never meet $\Rightarrow$ segregation
• A way to increase the number of independent observations with random matching

Absolute Stranger Matching

• Two participants never meet twice
• 
• PRO: Perfect control about reputation concerns
• CONS: Observations are not independent
• Everyone meets everyone

Stages of an Experiment

• Design
• The building blocks should meet research hypotheses
• Repetitions, Payment scheme, Choice task, Matching
• Useful to discuss the design in internal seminars
• Recruiting of participants
• Usually participants are registered in a database (e.g., ORSEE)
• Participation to the experiment is solicited via emails or posters
• Conductance of the experiment
• Participants are gathered in a room (laboratory) and asked to make their choices
• Computerized or paper & pencil
• Data analysis
• Data are collected, organized and analyzed
• Focus on hypotheses testing

State of the Art

Experimental Economics

• The number of publications referring to "Experimental Economics" has grown rapidly

Replications

• Experiments can be replicated (unlike field happenstance data)
• Most journals publish details about procedures and instructions
• Reproducing results is key to the credibility of the discipline
• Two studies present evidence of reproducibility of results
• In economics (Camerer et al, 2016)
• In psychology (Open Science Collaboration, 2015)

Replicability of Economic Experiments

• Camerer et al. (2016) autonomously replicate 18 experimental studies published in AER and QJE
• 
• Significant effect like the original in 11 out of 18 (61.1\%)

Replicability of Economic Experiments (ii)

• Camerer et al. (2016) compare their results to those of Open Science Collaboration (2015) in Pychology
• 
Evaluate 100 studies and replicate 36\% of them
• According to Camerer et al. (2016) the better relative performance of Economics may be due to
• Monetary incentives and truthful information
• Instructions and data are routinely published with the paper