What's the difference between time-series econometrics and panel data econometrics? This question may be very naive, but the way I'm taught econometrics I'm very confused if there's a difference between time-series and panel data method.
Regarding time series, I've covered topics such as covariance stationary, AR, MA, etc.
Regarding panel data, I've only seen discussions in the form of fixed effect vs random effect (or more generally, hierarchical model), difference-in-differences, etc.
Are these topics related in some ways? Since panel data also has a time dimension, why is there not discussion of AR, MA, etc. as well?
If the answer is that my education on panel methods is simply insufficient, could you point to a book that covers more than just FE/RE, difference-in-differences?
 A: The second dimension of panel data need not be time. We could have data on twins or siblings or data on N individuals answering T survey questions. Longitudinal data, where T is a second dimension, is arguably the most common type of panel data, and has become virtually synonymous with it.
Micro or short panels (large N, small T) typically have asymptotics that send N to infinity, keeping T fixed. Macro or long panels have moderate N and large T, and the asymptotics tend to hold N fixed and grow T, or grow both N and T. With micro panels, cross-unit dependence is typically not an issue because units are randomly sampled, whereas with macro panels it may be a real concern (spatial dependence between countries or states, for example). With macro panels, you also have to worry about unit roots, structural breaks, and cointegration, all of which are familiar time series concerns. You also have to occasionally worry about selectivity problems (like attrition, self-selectivity, and non-response). When T is long enough, even countries can disappear. 
I would take a look at Baltagi's Econometric Analysis of Panel Data, particularly chapters 8, 12, and 13. It also covers the short panels in some detail. The previous edition also had a companion volume with exercise solutions that was very nice.
A: As mentioned above then panel data has often been used on individual level rather than on an aggregated level with large N and small T. There are many pros with using panel data since we can remove individual heterogeneity and often get higher power when testing to mention two. This new time dimension does introduce some new methods, assumptions and problems compared with cross-sectional data (I will refer you to Wooldridge's book to study these closer). 
It is however very common within economics to also use country level panel data with small N and large T. This introduces a whole range of difficulties not encountered when dealing with large N, small T panel data. We could for instance have unit roots in our panel and there are also specific panel unit root tests to deal with this specific issue. Notice that these have a significantly higher power than unit root tests on individual series. We could also have all sorts of other kinds of non-stationarity in these panels. Furthermore, when dealing with panel data with small N and large T we can also have cointegration. Another major issue when dealing with large T and small N panel data is that this data is often for country level economic variables and that in this case the independence assumption is often violated and this should be tested for. That being said this is not a problem only for small N and large T but can also be present in large N and small T panels.
So panel data with large N and small T introduce a time series dimension compared to cross sectional data and are similar to cross sectional analysis while panels with large T and small N introduce a cross sectional dimension compared to the time series approach and which is similar to time series analysis.
An excellent book on panel data with large N and small T is "Econometric Analysis of Cross Section and Panel Data" by Wooldridge. This book is quite dense and packs a lot of information on every page so you might want to start with a introductory book in econometrics and read the section on panel data there first.
I do not know a specific book for panels with large T and small N but there is a volume called: "Nonstationary Panels, Panel Cointegration, and Dynamic Panels", Baltagi, ed. 
A: At least in the social sciences you often have panel data that has large N and small T asymptotics, meaning that you observe each entity for a relatively short period of time. This is why applied work with panel data is often somewhat less concerned with the time series component of the data.
Nevertheless time-series elements are still important in the treatment of panel data. For instance, the degree of auto-correlation determines whether fixed effects or first differences is more efficient. In difference in differences proper treatment of the standard errors to account for autocorrelation is important for correct inference (see Bertrand et al., 2004). Dynamic panels using estimators for small N, large T asymptotics are also available, you often find such data in macroeconomics. There you may run into known time-series issues like panel non-stationarity.
An excellent treatment of these topics is provided in Wooldridge (2010) "Econometric Analysis of Cross Section and Panel Data".
A: It's largely a question of emphasis, since both data consist of cross sectional and time series components.
Panel data is more likely to have large N and smaller T.
There is more attention to the individual components (e.g. stores over time, consumers over time) and more likelihood of segmenting those individual components (e.g. high income consumers, consumers who have moved from middle to high income).
The individual components have survival/replacement issues (the components leave the study for some reason, and must be replaced). With econometric data you are more likely to be dealing at a more aggregated level and it's often somebody else's problem (e.g. those fine folks at the BLS) to deal with those issues.
Autocorrelation issues do arise, but often are modeled as past history rather than as an autocorrelation per se, e.g. your past history of buying Chocolate Frosted Sugar Bombs http://www.gocomics.com/calvinandhobbes/1986/03/22 informs the prediction of future buying behavior. 
A: I would like to complement the above answers with a reference where you can read more about time dependence in panel data models, as you requested: Verbeek, Marno. A guide to modern econometrics, Wiley. 
There is a chapter in this book on panel data models that can serve as a good introduction.
As an example of contemporary research regarding time-dependence in panel data, you could read:
Fredrik N. G. Andersson: Exchange rates dynamics revisited: a panel data test of the fractional integration order. Empir Econ (2014) 47:389–409. 
