[This entry should be removed, see Applied-Econometrics.md]

Specification test

- Likelihood ratio (LR) test
- Heteroscedasticity, serial correlations, and nonnormality
- Nelson test, 1981
- Hausman test

- Individual-specific effect
- F test between the within and the pooling model
- Lagrange multiplier (LM) tests of the pooling model

- Fixed Effects vs Random Effects
- Hausman test [based on the difference of coefficient estimates]

- Serial Correlation
- Wooldridge’s test for “short” FE panels

Model selection

- AIC (Akaike information criterion)
- BIC (Bayesian information criterion)

Semi-parametric Regression

- Least absolute deviation (LAD) estimator
- Maximum score (MS) estimator; (Manski, 1975)
- Smoothed maximum score estimator; (Horowitz, 1992)

- Censored LAD estimator; (Powell, 1981, 1983)

- Maximum score (MS) estimator; (Manski, 1975)
- Symmetrically censored least square (SCLS) estimator; (Powell, 1986)
- Partially Linear Model
- Robinson Difference Estimator (Robinson, 1988) (9.7.3) (16.5)

- Single Index Models (9.7.4)
- Generalized Additive Models

Nonparametric Estimation

- Kernel Density Estimation
- Conditional Density Estimation
- Nonparametric Regression

Binary outcome models

- MLEs as latent variable models
- Linear probabilistic model
- Logit (logistic regression) model
- Probit model

- Grouped and Aggregated Data: Minimum chi-square estimator

Multinomial outcome models

- Unordered outcomes
- conditional logit (CL), multinomial logit (MNL)
- Independence of Irrelevant Alternatives

- nested logit (NL)?, three-level nested logit
- multinomial probit (MNP)

- conditional logit (CL), multinomial logit (MNL)
- Ordered outcomes, and sequential decision
- ordered logit, ordered probit

Choice-based sampling: weighted MLE

Sample Selection Models

- Tobit model (Censored model)
- MLE
- Two-step estimator (Ahn and Powell, 1993)

- Bivariate Sample Selection Model (Type 2 Tobit Model)
- Heckman two-step estimator (Heckman, 1979)

- Roy models (Type 5 Tobit Model)
- Simultaneous equations models (coherency condition)
- simultaneous equations Tobit/Probit model

Duration Regression Models:

- Proportional Hazard
- Left Censoring
- Markov Chain Models

Count Data Models:

- Poisson and Negative Binomial Models
- Simulated Maximum Likelihood (SML)

Short Panels (n << T)

Simple Regression Models with Variable Intercepts:

```
y_it = x_it β + z_i α + u_i + ε_it
dep = time-var + time-inv + individual + error
```

- Pooled Model (disregard time periods)
- Pooled OLS estimator [OLS over panel]

- Individual-specific Effects Model
- Random Effects (RE) Model (random intercept model) (equicorrelated model) (random components model) [individual-specific effect uncorrelated with regressors]
- Between Estimator [OLS over individual time-averages]
- Random Effects Estimator: [Feasible GLS over panel], [MLE]

- Fixed Effects (FE) Model [individual-specific effect correlated with regressors]
- Within Estimator (Fixed Effects Estimator) (Least-squares Dummy-variable (LSDV) Estimator) (Covariance Estimator) [OLS over panel after subtracting individual time-averages]
- First Differences (FD) Estimator [OLS over panel after first-differences in time]

- Random Effects (RE) Model (random intercept model) (equicorrelated model) (random components model) [individual-specific effect uncorrelated with regressors]

Individual effects are treated as nuisance parameters, while estimation of marginal effects are of sole interest.

Dynamic Models with Variable Intercepts: (AR(1))

```
y_it = γ y_it-1 + x_it β + z_i α + u_i + ε_it
dep = lag_dep + time-var + time-inv + individual + error
y_it = z_i α + x_it β + γ y_it-1 + u_i + ε_it
dep = time-inv + time-var + lag_dep + individual + error
```

- Random Effects Models
- General FGLS Estimator [1. OLS residuals for error covariance matrix estimation; 2. FGLS]
- GMM Estimator [GLS, lagged values as IV; in first differences]

- Fixed Effects Model
- General FGLS
- Fixed Effect GLS Estimator (FEGLS)
- First-difference GLS (FDGLS)

- GMM Estimator [GLS, lagged values as IV; in first differences]
- Transformed ML Estimator

- General FGLS

**General FGLS Estimator** is based on a two-step estimation process: first a model is estimated by OLS (pooling), fixed effects (within) or first differences (fd), then its residuals are used to estimate an error covariance matrix for use in a feasible-GLS analysis.
This framework allows the error covariance structure inside every group (individual time series) to be fully unrestricted and is therefore robust against any type of intragroup heteroskedasticity and serial correlation.
Conversely, this structure is assumed identical across groups and thus general FGLS estimation is inefficient under groupwise heteroskedasticity.
Efficiency requires N >> T.

**GMM Estimator**

```
y ~ covariates | gmm instruments | 'normal' instruments
```

By default, all the variables of the model which are not used as GMM instruments are used as normal instruments with the same lag structure as the one specified in the model. Transformation: difference GMM; system GMM.

first difference with IV estimator

**Pooled OLS** is biased upward and is inconsistent.
**GLS** and **ML** estimators are also generally biased.
**Within estimator** is biased, because eliminating the individual effect causes a correlation between the transformed error term and the transformed lagged dependent variable.

- Spatial Approach,
- Factor Approach,
- Cross-sectional Mean Augmented Approach,
- Test of Cross-Sectional Independence

**General FGLS** is inefficient under cross-sectional correlation.