What does a large t-statistic mean? I am empirically studying Fama & French three factors (market, SMB, HML - independent variables) model on a Viet Nam stock exchange. The dataset is daily stocks' return during almost 4 years. The stocks are grouped in six portfolios - each is dependent variable in each regression. I would like to test the validity of Fama & French three factor model in explaining excess return of portfolios. Below is regression output of one of the portfolios, with a really high t-statistics. Any problem with that?
            Coefficients    Standard Error        t Stat    P-value
Intercept       0.00191         0.00031          6.27072    0.00000
RPt = Rmt - Rf  0.98782         0.01737         56.85940    0.00000
SMB             0.87223         0.02796         31.19128    0.00000
HML             0.68983         0.02172         31.76355    0.00000

Please let me know if it's unclear.
 A: My Answer: Those numbers are probably okay. What is the basis for this assertion? I simply had a look through the source material:
Fama, E. F. and French, K. R. (1993) Common risk factors in the returns on stock and bonds, Journal of Financial Economics, 33, 3–56
In particular, have a look at table 6 (on p24). You'll see that the numbers they get are roughly similar to the numbers you are getting. Of course, it is impossible to say for sure given that you've only posted the results for one of your portfolios. I recommend going and seeing if the numbers for all your portfolios are "in the ball-park" of the numbers in the aforementioned table.
One other thing, I think you mean "uncorrelated" when you say "independent" regarding Market, SMB, HML.
My Rant: The regression based approach of Fama and French is really not the right way to go about tackling this problem. Essentially, in the estimation of the regression coefficients, you are assuming the model is correctly specified. But ultimately the specification of the model is the very thing you then try and test with the regression coefficients. A much better approach to this problem can be found in:
Bai, J., Ng, S. (2006) "Evaluating latent and observed factors in macroeconomics and finance", Journal of Econometrics 131, pp507-537.
Also, given that I've taken the time to write out this answer, I'm going to shamelessly plug a paper of my own on this subject :-)
Colin T. Bowers & Chris Heaton (2013): What does high-dimensional factor analysis tell us about risk factors in the Australian stock market?, Applied Economics, 45:11, 1395-1404
A: This is generally not a problem. It just means, for each coefficient, that if you assume it really is zero, and the effects you see in the estimated coefficient are simply due to sampling error (the randomness inherent in taking a random sample), your data are extremely unlikely. Your high t-statistic, which translates into a low p-value, simply says that something very unlikely has happened if your coefficients are zero in reality.
You can take the output of your regression as (statistical, that is, quantitative, not definitive) evidence of an effect (not causality, rather an association between the concerning independent and your dependent variable). Neither t-statistic nor p-value (which have a one-to-one correspondence here through the t distribution) make a statement about effect size. So you can be somewhat sure there is an effect. That's what the high t statistic means here.
