Business cycle stylized facts are not country specific? Business cycle and macroeconomic stylized facts are country-specific, right? But in my macro classes, they teach it as something universally true whiles using US data...:). For example, after an RBC model, we only check for US data where $\frac{C_t}{Y_t} &lt; 1$, end of story...:). How about other countries where $\frac{C_t}{Y_t} &gt; 1$? The model is not good for that country?
There are several aspects to this question:
General aspects:

What determines a stylized fact?
Are stylized facts country-specific?
Why do economic studies often use the United States as a benchmark?

Specific aspects:

What do we know about the macroeconomic quantity $C_t / Y_t$?

It will benefit our understanding to consider each of these questions separately.
1.
The term &quot;stylized fact&quot; is not very precisely defined in economics, and it is understandable that it confuses you. In my personal experience, most economists have an intuitive understanding of what constitutes a stylized fact, but would struggle to provide a precise definition when prompted.
Intuitively, stylized facts are the closest economics, and social science at large, can come to empirical predictions arising from scientific laws. The study of what precisely determines a scientific law constitutes a whole branch of philosophy of science by itself. Let me, for the sake of discussion, simplistically assume that scientific laws are statements about causation.
Since most social systems, such as the economy, are extremely complex, it is practically impossible to observe phenomena under controlled conditions. Any causal force that can be observed is usually counteracted by myriad other causal forces, and scientific experimentation aimed to isolate a single causal effect is usually ethically and practically out of bounds. As a result, it is exceedingly difficult to falsify economic &quot;laws&quot; (insofar as these exist) by way of precise empirical predictions, as is for example common practice in physics (see, e.g., the long history of attempts to falsify Einstein's general relativity theory). Stylized facts constitute a less precise measuring stick for the falsification of economic laws: they are robust empirical regularities observed in a certain social system, which any proposed causal mechanisms (in the context of a scientific law or model) characterizing that system have to generate as a baseline. In your case, if I write down a model of the macroeconomy, I typically want it to generate $C_t / Y_t &lt; 1$, unless I have a good reason otherwise -- that is, unless I can posit a causal mechanism that explains my deviation from that stylized fact.
The linked philosophy of science article provides a more formal discussion of the use of the term stylized fact in the social sciences. The author distinguishes four features of stylized facts, which they say involve (p.607-608):

Claims about the kinds of things that exist in the world: the categories that divide the world into an analytical ontology.
A way to operationalize those categories for analysis, usually in relation to a particular dataset or statistical classification (e.g. $C_t / Y_t$ only has meaning in the context of national accounting).
A claim of non-robust dependence (&quot;Y usually depends on X&quot;, &quot;$C_t / Y_t$ is usually smaller than 1&quot;).
A normative claim that the &quot;particular regularities identified are the ones most important to study and are preferable to other potential characterizations of the evidence&quot; (p.608). (A silly example: $C_t / Y_t$, the consumption share of output, tells us something important about an economy; the share of the value of cat food in output less so.)

2
Stylized facts are not supposed to be country-specific. Rather, they are meant to capture some general feature of an arbitrary economic system. However, due to the fact that economics often focuses on select sets of countries, and that stylized facts typically hold generally but not universally, it may appear differently. This brings us to...
3
It is true economics has traditionally focused excessively on advanced economies, and in particular the US. This has shifted dramatically in recent years, though. Still, a lot of economics teaching has not caught up with this shift in research focus, and it may certainly seem strange to students in countries / economies that are very different from the US that it is so often used as a benchmark.
This recent article in The Economist discusses this question in detail, and provides several reasons for economics's focus on particular countries:

The size of a country's economy accounts for 80% of variation in research attention.
The quality and availability of data.
A country's use of the English language.
Professional incentives: papers focusing on the United States have been published more frequently in top journals.

4
Now, to get to your question about $C_t / Y_t$. Though your other questions were interesting and important in their own right, they didn't really need to be answered: simple logical thinking could have already gotten you a long way.
$Y_t$ denotes income, $C_t$ consumption. Typically, it is not possible for a country to consume more than it earns, except through borrowing or transfers. Hence, as a baseline, $C_t / Y_t &lt; 1$. This is actually a very unsurprising empirical regularity, and not that interesting of its own right (as a matter of fact, the deviations from the baseline are much more interesting). See the data here and here (depending on your definition of C_t) to discover which countries constitute the exception to the rule.