How are models equations defined in this paper? In this paper, The Mediating Role of Perceived Value: Team Identification and Purchase
Intention of Team-Licensed Apparel,
three different Models (A,B and C respectively) are analyzed based on a survey with 9 questions about team identification (3 ordered categorical variables), perceived value (3 ordered categorical variables) and purchase intention (3 ordered categorical variables). I have difficulty to understand which are the actual dependent and independent variables of each Model. For example, Model A may be defined as:
$$
\textrm{PurchaseIntention} = \alpha + \beta_1\textrm{TeamID}+\beta_2\textrm{PerceivedValue}
$$
but how to deal with three different proxies for PurchaseIntention and for TeamID and PerceivedValue? Model B and Model C are even unclear.
 A: 
I have difficulty to understand which are the actual dependent and independent variables of each Model.

The terms independent and dependent can be tricky because variables are often not independent and dependent (see the many responses to the question In Regression Analysis, why do we call independent variables "independent"? which advocate the use of different terms like predictor and response variables).
These terms 'dependent' and 'independent' relate to controlled studies where an experimenter is able to control the values of certain variables and these values are then considered independent of other variables and random fluctuations in the experiment (independent from anything except for the choices made by the experimenter). Other variables in the experiment are observed and the experimenter wishes to determine how their values depend on the change that the experimenter makes in the controlled (independent) variables.
In the study that you cite, they don't control variables and it is not exactly right to speak about dependent and independent variables (there's no input/output, everything is uncontrolled). However, they seem to consider/assume a causal model where purchaseIntention is a function of (dependent on) other variables TeamID and PerceivedValue.

*

*'purchase intention' is a response or outcome variable (and some call this dependent).


*'team id' is a predictor variable (and some call this independent).


*'perceived value' is considered differently in the different models, either as 'mediator' or as 'predictor'.
Regarding the latter variable, the degree in which it is a mediator or predictor dependends on the degree by which the mediator ('perceived value') depends on the predictor ('team id'). See
Baron, Reuben M., and David A. Kenny. "The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations." Journal of personality and social psychology 51.6 (1986): 1173.


A variable functions as a mediator when it meets the following conditions: (a) variations in levels of the independent variable significantly account for variations in the presumed media-tor (i.e., Path a), (b) variations in the mediator significantly account for variations in the dependent variable (i.e., Path b), and
(c) when Paths a and b are controlled, a previously significant
relation between the independent and dependent variables is no
longer significant, with the strongest demonstration of mediation occurring when Path c is zero.

The article that you cited is testing conditions (a) and (b) but is not a controlled experiment and doesn't test condition (c).

2

but how to deal with three different proxies

In the subsection 'instrument' of the section 'method' of the article, they describe the use of 7 point scales. They do not say that they simply considered this scale as a numerical variable and convert the different items into an average, but you can assume this based on the reported results all being values between 1 and 7. There are several references in that section to articles from which they adopted the scales, in those articles you might read whether a simple averaging is used or something more fancy (It is almost always the simple approach instead of something complex).
