Difference between one-way and two-way fixed effects, and their estimation

Question

Consider a basic linear unobserved effect panel data model, e.g.: $$Y_{it}=\beta x'_{it}+c_i+\lambda_t+u_{it}, \quad t=1,\dots,T$$ where the vector $x_{it}$ contains the independent variables and $u_{it}$ is an error term. Number of individuals is $N$. Assume that the the unobserved individual effect $c_i$ may be correlated with $x_{it}$ (fixed effects assumption).

First question: What is the difference between a "one-way" and a "two-way" model? Wooldridge (2006 & 2010) never uses this terms. I assume a one-way model does not include time effects, e.g.: \begin{equation}Y_{it}=\beta x'_{it}+c_i+u_{it}, \quad t=1,\dots,T \quad(1)\end{equation} but a two-way model additionally does include time effects : \begin{equation} Y_{it}=\beta x'_{it}+c_i+\lambda_t+u_{it}, \quad t=1,\dots,T \quad(2) \end{equation}

According to the Frisch-Waugh-Lovell theorem the within estimator and the least squares dummy variable (LSDV) estimator both yield the same coefficients for equation (1).
This leads me to my second question: How would you estimate equation (2)? Can you first include $T$ time dummies (should it be $T-1$?) and then choose between the within estimator or the LSDV estimator? This answer suggests to first use the within transformation and afterwards to estimate the model including dummy variables, what I find confusing.

References:
Wooldridge, J. M. (2006). Introductory econometrics (3rd ed.). Thomson/South-Western.
Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data (2nd ed.). The MIT Press.

EDIT: I had a look at some lecture notes (here, here and here) and it seems like two-way models include what are sometimes called "time (fixed) effects" (see $\lambda _t$ in equation (2)), as I assumed.
What I understood was that either a) a somewhat different within transformation can be applied to two-way models, or b) dummies are included for one dimension (either time or individual) and then the "normal" within transformation (subtracting means) for the other dimension is applied.
Depending on estimation procedure the standard errors have to be corrected because of different degrees of freedom. Also, procedure a) seems only to be valid for balanced panels. I haven't found a textbook reference yet.

Answer 1

The unobserved effects model is modeled as: \begin{equation} y = X\beta + u \end{equation} where \begin{equation} u = c_{i} + \lambda_{t} + v_{it} \end{equation}

A one-way error model assumes $\lambda_{t} = 0$ while a two-way error allows for $\lambda \in \mathbb{R}$ and that is the answer to the first question.

The second question cannot be answered without more assumptions about the error structure or purpose of the study. Using Wooldridge (2010) chapters 10 and 11, generalize each of the assumptions to cover the temporal error structure as well. For example, when considering POLS, the critical assumption is $\mathop{\mathbb{E}}\left(\mathbf{x}_{it}^{\prime}u\right) = 0$. In the chapter it is summarized as meeting the following conditions:

1. $\mathop{\mathbb{E}}\left(\mathbf{x}_{it}^{\prime}c\right) = 0$
2. $\mathop{\mathbb{E}}\left(\mathbf{x}_{it}^{\prime}v\right) = 0$

However, if one does not assume $\lambda_{t} = 0$, i.e., two-way error model, a third condition must be satisfied for consistency of the POLS estimator: \begin{equation} \mathop{\mathbb{E}}\left(\mathbf{x}_{it}^{\prime}\lambda\right) = 0 \end{equation} and so on.

In the case of estimating the fixed effects, one can go with LSDV (including indicators for the panel ID and temporal ID), but the dimension might become unfeasible fast. One alternative is to use the one-way error within estimator and include the time dummies such as one usually do with software that does not allow for two-way error models like Stata. A third and most efficient way is to estimate it with the two-way error within estimator. \begin{equation} y_{it} − \bar{y}_{i.} − \bar{y}_{.t} + \bar{y}_{..} = (x_{it} − \bar{x}_{i.} − \bar{x}_{.t} + \bar{x}_{..})\beta \end{equation} This approach is coded in several statistical packages such as the R package plm and correctly adjust the degrees of freedom to include the T - 1 additional parameters compared to the one-way error within estimator. Most two-error way model estimators are not limited to balanced panels (only a handful). For short-panels running the one-way error within estimator with time dummies is feasible. As a side note, even if one gets the estimates for the temporal effects it is important to notice that as with the LSDV fixed effects for one-way error models these are not consistent as the estimates increase in number and length of panels.

I recommend Baltagi (2013) textbook for a pretty comprehensive explanation of the estimators for one-way and two-way error models.

References:

Baltagi, Badi H. 2013. Econometric analysis of panel data. Fifth Edition. Chichester, West Sussex: John Wiley & Sons, Inc. isbn: 978-1-118-67232-7.

Croissant, Yves, and Giovanni Millo. 2008. “Panel Data Econometrics in R : The plm Package.” Journal of Statistical Software 27 (2). doi:10.18637/jss.v027.i02.

StataCorp. 2017. Stata 15 Base Reference Manual. College Station, TX: Stata Press.

Wooldridge, Jeffrey M. 2010. Econometric Analysis of Cross Section and Panel Data. Kindle Edition. The MIT Press. ISBN: 978-0-262-23258-8.

Answer 2

From this paper on reliability and intra-class correlation coefficient: https://www.sciencedirect.com/science/article/pii/S1556370716000158

One-Way Random-Effects Model In this model, each subject is rated by a different set of raters who were randomly chosen from a larger population of possible raters. Practically, this model is rarely used in clinical reliability analysis because majority of the reliability studies typically involve the same set of raters to measure all subjects. An exception would be multicenter studies for which the physical distance between centers prohibits the same set of raters to rate all subjects. Under such circumstance, one set of raters may assess a subgroup of subjects in one center and another set of raters may assess a subgroup of subjects in another center, and hence, 1-way random-effects model should be used in this case.

Two-Way Random-Effects Model If we randomly select our raters from a larger population of raters with similar characteristics, 2-way random-effects model is the model of choice. In other words, we choose 2-way random-effects model if we plan to generalize our reliability results to any raters who possess the same characteristics as the selected raters in the reliability study. This model is appropriate for evaluating rater-based clinical assessment methods (eg, passive range of motion) that are designed for routine clinical use by any clinicians with specific characteristics (eg, years of experience) as stated in the reliability study.

Two-Way Mixed-Effects Model We should use the 2-way mixed-effects model if the selected raters are the only raters of interest. With this model, the results only represent the reliability of the specific raters involved in the reliability experiment. They cannot be generalized to other raters even if those raters have similar characteristics as the selected raters in the reliability experiment. As a result, 2-way mixed-effects model is less commonly used in interrater reliability analysis.