Intermediate Econometrics

• Some regressors may be endogenous and this creates:

  • Biased estimates
  • Inconsistent coefficient estimates
  • Wrong causal interpretation

• Sources (Omitted variables, Reverse causality, Measurement error, Selection bias)

• In linear models the two Stage Least Square works

• In binary models, nonlinearity requires special methods

• We focus exclusively on the IV-Probit framework, because the probit model is:
  • Mathematically convenient
  • Fully compatible with IV methods
  • The standard reference in the econometrics literature
• A IV estimator for logit:
  • Is not standard
  • Requires complex structural assumptions
  • Less commonly used in core applied research

• There exists an unobserved latent variable
• We only observe:
  • 1 = decision taken
  • 0 = decision not taken
• In formulas, we observe a binary outcome, \(y_i\), while the latent variable \(y^*_i\) represents the underlying economic decision

\[ y_i = \begin{cases} 1 & \text{if } y_i^* > 0 \\ 0 & \text{if } y_i^* \le 0 \end{cases}\]

• The model is \[y_i^* = x^\top_i \beta + \varepsilon_i\]

• We split \(x^\top_i\) into two parts :

  • \(w_{i}\): exogenous regressors
    
  • \(z_i\) = endogenous regressors

• While \(\varepsilon_i\) is the structural error

• This leads the model to be written as \[y_i^* = w^\top_i \alpha + z^\top_i \gamma + \varepsilon_i\]

• Each endogenous regressor is generated by: \[z_{gi} = \pi_g^\top u_i + \nu_{gi}\]
  • \(u_i = (w_{i}, r_{i})\)
  • \(r_{i}\) = external instruments
  • \(\nu_{gi}\) = first-stage error
• Endogeneity is captured by correlation between the errors:

\[\text{Cov}(\varepsilon_i , \nu_i) \neq 0\]

• This implies:
  • Standard probit is inconsistent
  • Instruments are needed to break this correlation

• Under joint normality of \((\varepsilon_i, \nu_i)^\top\), the conditional mean of the latent outcome is:

\[\mathbb{E}(y_i^* \mid w_{i}, z_i, u_i) = \gamma^\top x_i + \rho^\top \nu_i\] - \(\rho\) measures the strength of endogeneity

• This leads to the Control Function Probit: \[P(Y_i = 1 \mid w_{i}, z_i, \hat{\nu}_i) = \Phi(\gamma^\top x_i + \rho^\top \hat{\nu}_i)\]

  • \(\hat{\nu}_i\) are residuals from the first stage
    
  • \(\rho\) captures endogeneity
    

• Thus, we can test: \[ H_0 : \rho = 0\]

  • If rejected → endogenous regressor is truly endogenous
    
  • If not rejected → standard probit is sufficient

• This is the Rivers–Vuong Two-Step IV-Probit Estimator

• Within the probit framework, we can use the following approaches:

1. Maximum Likelihood (ML-IV Probit)
2. Two-Step / Control Function (Rivers–Vuong)
3. Minimum Chi-Square (Newey)

• These are the standard IV estimators for probit models with endogeneity

• Pros
  • Highly efficient
  • Best statistical properties
• Cons
  • Very complex
  • Many parameters
  • Sensitive to misspecification
• Used mainly in advanced research.

• Step 1
  • Regress endogenous variables on instruments
  • Save the residuals
• Step 2: Run probit using:
  • Original regressors
  • Plus residuals
• Key Intuition
  • Residuals capture endogeneity
  • If they matter, then endogeneity exists
• Pros:
  • Direct economic intuition
    
  • Easy to implement
    
  • Allows explicit testing for endogeneity
    
  • Same logic as 2SLS in linear models
• Cons: Slightly less efficient than full ML
  

• Combines several intermediate estimates

• Chooses parameters that minimize statistical distance

• Pros

  • More efficient than Two-Step

• Cons:

  • Highly technical
    
  • Rare in empirical practice
    

• Research Question: What drives banks to use FX derivatives?

• Dependent Variable: Use of derivatives: Yes / No

• Key Explanatory Variables

  • Managerial ownership
  • Bonus
  • Stock options
  • Financial leverage

• This example is from Adkins, Carter, and Simpson (2007) and Adkins (2012)

• These authors analyzed the effect of managerial incentives on the use of foreign-exchange derivatives for hedging by U.S. bank holding companies, for the 1996-2000 period.

• The dependent variable federiv is 1 if the bank uses foreign-exchange derivatives.

• The first set of covariates concerns Manager ownership. When managers have a higher ownership position in the bank, their behavior is more in line with the preferences of shareholders, and they therefore have an incentive to take risk

  • The logarithm of the percentage of total shares outstanding that are owned by officers and directors (linsown) should therefore have a negative effect on the probability of using foreign-exchange derivatives. However, incentives provided by regulation may dominate the expected incentive relation and lead to a negative effect on the probability
  • On the contrary, institutional blockholders have imperfect information and, therefore, the logarithm of the percentage of total shares outstanding that are owned by all institutional investors (linstown) should have a negative effect on the probability of using foreign-exchange derivatives

• The second set of covariates concerns CEO compensation:
  • Value of option awards (optval) should induce managers to take more risk and therefore should have a negative effect on the probability
  • On the contrary, cash bonus (bonus) may increase the probability of hedging in order to decrease variability in the firm’s cash flows
• The other covariates are:
  • the leverage (eqrat)
  • the logarithm of total assets (ltass)
  • the return on equity (roe)
  • the market to book ratio (mktbt)
  • the foreign to total interest income ratio (perfor)
  • a derivative dealer activity dummy (dealdum)
  • dividends paid (div) and the year from 1996 to 2000 (year).

• Compensation affects risk-taking

• Risk strategy also affects compensation

• Leverage is jointly determined with risk

• Three covariates are suspected to be endogenous:

  • the leverage (eqrat)
  • the option awards (optval)
  • the bonus, (bonus)

• The external instruments are:

  • the number of employees (no_emp)
  • the number of subsidiaries (no_subs)
  • the number of officies (no_off)
  • the CEO age (ceo_age)
  • the 12 month maturity mismatch (gap)
  • the ratio of cash flow to total assets (cfa)

• These affect incentives but not directly derivative use.

• To fit the models we use binomreg function of the micsr package

• The coefficients of linstown, bonus and optval have the expected sign. linsown has a positive sign, which must be driven by the strength of the regulatory constraints
  • Bonuses → Increase hedging
    
  • Stock options → Increase risk-taking → Less hedging
    
  • Institutional ownership → Less hedging
    
  • Managerial ownership → Effect reversed due to regulation
    

• A Wald test that \(\rho = 0\) can be performed more simply using the miscsr::endogtest function:
• Endogeneity is weak, but statistically significant at \(10\%\)

• The fundamental problem of causal inference with observational data is that:
  • Treated and control units not come from the same population
    
  • As a result:
    • Observable characteristics differ
      
    • Unobservable characteristics may also differ
• This leads to selection bias in treatment effect estimation.

• The idea of matching is:
  • For each treated observation, find a control observation that is as similar as possible, based on observable characteristics
• If matching is successful, the resulting sample:
  • Mimics a randomized experiment
    
  • Allows causal interpretation under selection on observables

• However, with many continuous covariates, exact matching is impossible.

• Solution: Propensity Score Matching

• Propensity score is: \[p(x) = P(T=1 \mid x)\]

  • \(T\) is treatment
  • \(x\) are the covariates

• It is estimated using Logit or Probit, where the treatment indicator as dependent variable

• Each treated unit is matched with the control unit having the closest propensity score.

1. Estimate the propensity score using a rich model
   
2. Divide observations into strata (e.g. 5 blocks)
   
3. Test equality of mean scores within each stratum
   
4. If the test fails → split the stratum
   
5. Test balance for all covariates
   
6. If balance fails → re-estimate a more flexible model

• Achieve covariate balance between treated and control groups

• Once strata are defined, the treatment effect is estimated as:

\[\sum_k (\bar{y}_k^T - \bar{y}_k^C) \, f_k\]

- $\bar{y}_k^T$ = mean outcome of treated units in stratum $k$
- $\bar{y}_k^C$ = mean outcome of control units in stratum $k$
- $f_k = n_k / n_T$: frequency of each group

• This gives the Average Treatment Effect on the Treated (ATT)

• Study of Ichino, Mealli, and Nannicini (2008)

• Goal: Examine the effect of temporary work agency (TWA) jobs on the probability of finding a stable job

• The data set, called twa, contains \(2030\) observations (\(511\) treated and \(1519\) untreated) for two regions, Tuscany and Sicily

• Let’s focus to the Tuscany obtaining a sample where group is the treatment variable

• The outcome is also a factor indicating the employment status one year after the program. Its levels are none (no job), other, fterm for fixed-term contract and perm for a permanent contract. Following the authors, we define the outcome of interest as a dummy for a permanent contract
• Outcome: Permanent contract after 1 year (perm)
• To get a first idea of the treatment effect, we compute the mean of the outcome for the two groups
• The proportion of individuals who have a permanent job is 31.3% for the treated group and 16.6% for the control group and the apparent treatment effect is therefore 14.7%

• The outcome is also a factor indicating the employment status one year after the program. Its levels are none (no job), other, fterm for fixed-term contract and perm for a permanent contract. Following the authors, we define the outcome of interest as a dummy for a permanent contract
• Outcome: Permanent contract after 1 year (perm)
• To get a first idea of the treatment effect, we compute the mean of the outcome for the two groups
• The proportion of individuals who have a permanent job is 31.3% for the treated group and 16.6% for the control group and the apparent treatment effect is therefore 14.7%

• The micsr::pscore function implements the algorithm previously described. The first two arguments are:
  • formula: it should have two variables on the left-hand side, the first indicating the outcome and the second the group (the group variable can be either a dummy or a factor with two levels, the second indicating the treated individuals)
  • data
• To use the same formula of Ichino, Mealli, and Nannicini (2008) we need to consider
  • square term for the distance to the next agency
  • an interaction between self-employed status and the city of Livorno

• Three supplementary arguments of pscore can be used
  • the maximum number of iterations (default 4)
  • the tolerance level for the t-tests (default 0.005)
  • the link for the binomial model (default to logit).
• pscore returns a pscore object which contains three tibbles:
  1. strata contains information about the stratas (the frequencies, the average propensity scores and the probability value of the hypothesis of no difference of the propensity scores in the two groups)
  2. cov_balance has a line for every covariate and contains the strata for which the probability value is the smallest
  3. model contains the original data sets with some supplementary columns:
  • pscore contains the propensity score for every observation
  • .gp is a factor with levels "control" and "treated"
  • .cs is a boolean indicating whether the propensity score for an observation lies in the interval of scores for the treated
  • .resp contains the response
  • .cls indicates the strata for the observation

• A summary method is provided and the print method for summary.pscore objects has a step argument that allows to print the result of each step of the estimator.

• Two of the initial stratas were cut in halves (0-0.2 and 0.6-0.8). The control subsample is restricted to the range of the values of the propensity score for the treated; therefore, only 592 observations of the control group out of 628 are used.

• With step = "covariates", we get a (long) table indicating, for each covariate, the results of the balance test between the treatment and the control group

• The values of the estimated ATET (average treatment effect of the treated)
  • The estimated treatment effect is \(0.177\), which is slightly higher than the treatment effect computed with the whole sample which was 14.7%
  • It is highly significant, being the standard deviation \(0.035\)

• An alternative to using strata and computing the ATET as a weighted average of the difference of the mean of the outcome between treated and control observations in each stratum is

  • to match each treated observation to one or several observations in the control group

• The simplest algorithm consists of selecting, for every treated observation, the control observation which has the closest value of propensity score

• Using the model tibble returned by pscore, we begin with constructing two tibbles, one for the treatment group and one for the control group, containing only the index of the observation and the value of the propensity score

• We then need to join the two tables
  • inequality: one can for example match a treated observation with all the observations in the control group with higher propensity scores
  • rolling: select only one observation, the closest one
• Inequality: We do it below only for the first two treated observations and they are matched to respectively 309 and 331 observations in the control group

• We then need to join the two tables
  • inequality: one can for example match a treated observation with all the observations in the control group with higher propensity scores
  • rolling: select only one observation, the closest one
• Rolling: this time one row is returned for every treated observation, and the same control observation can be matched with several treated observations

• Next, we match with the closest lower propensity score control:

• We then join the two tables (by the index for treated observations id_tr) and select the control observation which is the closest:

• For a couple of treated observations, the propensity score is greater than the highest propensity score in the control group
• Therefore, ps_sup is NA, and we set it to an arbitrary high value so that the id_inf observation is selected
• We can then compute the number of observations in the control group that are used and select control observations which are the most often used to match treated observations (for example, observation 4935 in the control group matches 16 observations in the treatment group)

• For some observations, the algorithm may result in a poor match for some treated observations if the difference between the probability scores of this observation and the matched control observation is high.
• In our sample, the highest difference is about \(2.6\)%.

• The sample can be reduced to observations for which the difference is lower than a given value: caliper matching.
• For example, to restrict the sample to treated observations for which the propensity score difference with its matched control observation is lower than 1%
• We then lose 25 treated observations

• To compute the treatment effect, we pivot the tibble in “long” format (one line for each observation) and we join it to tuscany to get the response (perm) for each observation

• The ATET (0.1886) is close to the one obtained previously using the algorithm based on stratas (0.1769)

• MatchIt implements advanced matching techniques
• Main function: matchit()
• Interface: formula + data
• Key argument: replace = TRUE → allows a control unit to match multiple treated units
• Default method: “nearest”

• The number of matched observations is 427 (as previously), the 281 treated observations and 146 observations of the control group

• A match.data function is provided in order to extract the data frame restricted to the treated observations and the subset of observations of the control group that match
• weights → for subsequent analyses
• Treated: weight = 1
• Controls: proportional to the number of matches (For an observation of the control group that matches only one treated observation, the weight is \(146 / 281 = 0.52\) and, for example, for an observation of the control group that matches four treated observations, the weight is :\(146 / 281 \times 4 = 2.08\).)

• Caliper matching is performed using the caliper argument.

• Once the matching has been performed, the quality of the balancing process can be assessed
• The plot method draws a Love plot; for every covariate, two standardized mean differences are plotted, one for the raw data set and one for the balanced one