As far as I understand, you want to keep the dependent variable - Abnormal Returns and, in the same time, introduce the same variable as an explanatory variable and perform the PCA. Although it's rather hard to discuss without the data at hand, I have two main concerns on the proposed methodology:

1. Even if you perform the PCA, the final result of the results of the regression will be biased due to auto-correlation. This is due to the fact that one of your explanatory variables will have a correlation of 1 with the dependent variable.

2. After you perform the PCA and "aggregate" the dependent variables, the number of the resulted variables may go down to 2 or 3. In this conditions, for a meaningful interpretation of the results, you will have to find a proper economic meaning of each of this variables and explain its influence on the dependent variable. For example, it might be difficult to find such economic rationale for a variable that combines Abnormal Return and Serial Buyer or Rationale.

Dear Anatol, 

First of all thank you so much for your answer.

I see your points and actually my major concern is bout point 1, that is, the theoretical rightness of including dEpenddent Variable into my model. Because of PCA does not really consider dependent variable like in Linear  Regression, Abnormal Returns should be treated as any other variable. In fact, the aim of my final section of thesis is to demonstrate how variables behave not only in function of Dependent variable, but also among them. 

Accounting to your second point, actually, the combinations I've found make sense aso comparing to the previous regressional model (obviously) as AR goes positive when also Cash increse and the transaction is conducted Cross Border. All these effects are in line with my initial hypothesis. In my view, it would be hard to prove these results without including AR in the PCA. 

Nick

Dear Nicola, I would not include the dependent variable in the PCA model, as doing so would create an additional and confusing correlation between the components and the dependent variable itself. Usually one would want to perform a PCA to reduce the number of covariates, and then regressing a dependent variable upon the components which have been found.

The principal component analysis (PCA) is a statistical technique that seeks to synthesize the information. The number of variables is reduced to once known what explains the largest percentage of the total variability of the system under study. The main components that form are a linear combination of the original variables and are mutually independent.

The ACP does not require compliance with the assumptions of normality nor homoskedasticity.

The only requirements are related to the continuity of the variables and the number of observations exceeds the number of variables.

So should not I introduce Dummy variables?

I presume that you wish to explain the dependent variable as a function of the independent variables. IN that case, don't include the dependent variable in the PCA. Rather, regress it on the components resulting from the PCA. As a result you will get regression coefficients describing the (linear) relation of the components to the dependent variable and R-squared and the components' individual contributions to it, i.e., how well the components explain (or "predict") the dependent variable.

If you manage to find

few components (k), able to explain most of the variance, may

replace the primitive variables with minimal loss of information. Thus

It will be available a few dummies, being few, contain all

original. This is the aim of principal component analysis: simplification

or reducing the initial table, NXP to nxk. Of course, if instead of selecting k

main components, the possible p were taken, there would be a loss of

information, but he would not have managed to simplify the problem.

Dear Nicola,

I take this contact to invite him to participate in meetings of biostatistics, econometrics and computing the V International Congress on Animal Production to be held in the Palace of Conventions in Havana, Cuba from 16 to 20 November this year. in this event some related multivariate statistics I see is your line of work papers will be presented. Please, if you are interested in this event, you can write to my email for sending all the necessary information. I hope that motivates and we can meet in Cuba. Best regards

My personal email: [email protected]

Dr. C. Raul Cobo Wedge

Researcher, ICA

I agree with all. If you include your dependant variable in the PCA,  any  analysis using its result will be fallacious.  Also, in this case, you will face a problem of heterocedasticity in the estimation of your econometric model.

Got it, the fact is that for PCA, I do not intend to re-use components for a new regression. Rather is a per-se analysis in which I just want to find evidence on what variables move alike and which ones move along with AR. For this I can't undestand ho to get the the latter group without including AR. Also online tutorial include AR when considering PCA as a simple per-se study.

Dear Nicola, 

I agre with the answer of Meltem and the meaning of principal componentes can help you in understanding why you should not include the dependent variable.

Remember that wiht PCA we try to use less explanatory variables, to simplify the analysis. Sometimes we aplly PCA to the independent variables  , for not having multiculionarity problems, and after that we aplly regression model where the Vi (= nº components choosen) are the VI needed to explain the VD.

...

PCA must have at leat 2 variables. For the result to be reliable the sample must  have at leat 5 times the nº of variables. Each component is a linear function of the original variables (similarr to linear regression except that there is no interecept term).

The first principal component maximize the amoutn of variance that is explained . Formally the linear function or principal component  is referred as eigenvector . The amount of total variance that is explained by an eigenvector is knoen as eignenvalue.

The second eigenvector would maximize the amoutn of remaining variance that was explained.etc.

The nº of eigenvectors required to explained 100% of total vriance is know as the rank of the correlation matrix, which is used to summarize the interrelationships among the original variables .

Have a nice time

Helena

PCA is a kind of unsupervise learning, meaning no independent variable in this analysis. 

Nicola,

PCA squeezes the variability of a set of variables into a factor. By inserting the dependent variable you will loose any sort of cause-effect relationship. If I have to say what is not on a text-book  I would suggest not to use PCA to confirm direct relationships from linear regression but an accurate selection of variables referring to theory and literature.

PCA gives the proportion of total variability in the dependent variable that is explained by a combination of independent variables. This is basically because it is a technique used in selecting variables that contribute significantly to explaining the dependent variable. The dependent variable should therefore, not be included in the when running a  PCA.

Dear Nicola, 

I agre with the answer of Meltem and the meaning of principal componentes can help you in understanding why you should not include the dependent variable.

Remember that wiht PCA we try to use less explanatory variables, to simplify the analysis. Sometimes we aplly PCA to the independent variables  , for not having multiculionarity problems, and after that we aplly regression model where the Vi (= nº components choosen) are the VI needed to explain the VD.

Should I insert dependent variable into Principal Component Analysis? - ResearchGate. Available from: https://www.researchgate.net/post/Should_I_insert_dependent_variable_into_Principal_Component_Analysis#view=55da7b276143257f998b45a5 [accessed Aug 24, 2015].

PCA is not the right method to search for evidence of co-movement, if I understood your problem. PCA seeks to find which factors contribute to the best answer to a question. If you include the question as an answer to itself, you are taking a path to nowhere, as most of the answers seem to agree.

costos predeterminados

PCA is a technique to account for the variability of the system from the linear combination of independent variables, thus it should not include dependent variables.

Thanks to everyone.

Without ignoring the many contributions that have arisen answering your question, let me suggest a way of analysis that has not been mentioned so far, except inattention on my own.

Basically, your idea of inserting a dependent variable into PCA sounds very much like the univariate regression PLS (Partial Least Squares).

PLS  is a projective exploratory technique, especially used in chemiometry for highly correlated regressors and in large number with respect to observations (cf. Svante Wold, 1983).

Essentially, the  NIPALS PCA-algorithm (Hermann Wold, 1966), on which the PLS regression is based, can be described as a covariance maximization process between the Y table to explain (in your univariate case reduced to an endogenous variable) and the  X explaining  table (your list of regressors or exogenous variables) .

In other terms, the PLS pseudo-factors extracted from the X table are the most linked by covariation to those extracted from the Y table (which is the dependent variable itself in the unvariate PLS, or PLS 1).

For a complete overview, see Wold & al, 2001:

http://www.libpls.net/publication/PLS_basic_2001.pd

For a simple introduction to the PLS regression with some examples in SAS, you can consult:

http://support.sas.com/rnd/app/stat/papers/plsex.pdf

For the R users, you can use the "pls" package, please see :

http://www.jstatsoft.org/v18/i02/paper?ref=driverlayer.com/web

and for the SPSS ones :

http://www.unt.edu/rss/class/Jon/SPSS_SC/Manuals/v19/IBM%20SPSS%20Statistics%20Base%2019.pdf

But, obviously, SIMCA-P (Umetrics), originally designed by Nouna Kettaneh-Wold,  provides the best functionalities :

http://www.umetrics.com/sites/default/files/kb/multivariate_faq.pdf

http://support.sas.com/rnd/app/stat/papers/plsex.pdf

http://www.jstatsoft.org/v18/i02/paper?ref=driverlayer.com/web

http://www.unt.edu/rss/class/Jon/SPSS_SC/Manuals/v19/IBM%20SPSS%20Statistics%20Base%2019.pdf

http://www.libpls.net/publication/PLS_basic_2001.pdf

http://www.umetrics.com/sites/default/files/kb/multivariate_faq.pdf

I agre with the answer of Meltem