I would look at Leslie Kish "Sampling Survey," New York: Wiley. (Kish 1965, p. 46); and George W., "Statistical Method," Fifth Edition, (Ames Iowa: The Iowa State College Press, 1959), p. 498 for classical sample sample formula.

For regression analysis. I find A. Sande, “A Sample Size Formula for Multiple Regression Studies,” Public Opinion Quarterly, Vol. 50, Spring 1986 very operational.

Clive Granger: "Empirical Modeling in Economics" (Cambridge Press, 1999) addressed panel samples. He wrote:, "most important macroeconomics series are rather short", and he would do a "post-sample" evaluation for time-series, a "cross-validation technique" for cross-section data, and an mixture of both for panel data. (Ibid., pp. 65-66).

ISSUE: Minimum sample size determination

DATA: The data here are gathered from 5 sources with 9 observations each. There are total of 45 individual observations. Can these observations be combined and treated as a single group of observations? We should not combine apple and orange. It would be helpful to verify if the 45 observations are homogeneous. Assume that the 5 firms are called A, B, C, D and E. Each firm with observations: XAi, XBi, …, XEi. The possible pairs are:

AB, AC, AD, AE

BC, BD, BE

CD, CE

DE

There are 10 possible pairs to compare these 5 firms. Finding these possible pairs may be given by:

(1)   Pairsmax = n(df) / 2

In this case, there are 5 companies or n = 5 and df = n - 1 or df =4; thus 594)/ 2 = 10. There are ten possible pairs of of the companies. Are these firms similar or different from one another? if they are homogeneous, then combining the 45 observation would make sense. Since the length of the data string is equal for all firms, i.e. ni = 5. Let's just use d-bar test.

(2)   td = d^ / (S / sqrt(n))

... where d^ = (d1 + d2 + ... + d5) / 5 ; S = standard deviation of XAi, ... etc. Assume that td is not significant, the data may be said to be homogeneous. we could now have combined observations of n = 45. From which population and what is the size of that population? we do not know. If N is non-finite, then the sample size is determined by;

(3)   n = (Z2σ2) / E2

... where Z = critical Z at a specified level of confidence interval; σ = estimated standard deviation; and E = standard error, i.e. σ / sqrt(n). This is under the assumption that with 45 observation, the characteristic of central limit theorem would be manifested. If not, this equation would not work because it assumes normal distribution. What happens if these 5 firms all have different distribution type? Can we combine them? We need to verify. See link for distribution gallery at NIST. 

One method used in health science based the sample size calculation on Number Needed to Treat (NNT) approach. The rationale is to provide an answer to "how many patient to treat in order to maintain the same level of survival rate?" The formula is:

(4)   NNT = 1 / C - T

... where C = control group and T = treatment group statistics. In the present case, if the test for distribution of the  45 observations shows normality, we may use 45 observations as the treatment group and use the theoretical value for the control group. To apply NNT to economics case, one would have to refine and redefine terms----no citation works here. Still speculative as a potential idea. See article on NNT.

DATA TYPE: Does it matter if the data is quantitative, ordinal or nominal. Agresti seems to say that that does not change the fact that minimum sample size of about 30 is the standard practice. see attached article by Agresti.

MULTIPLE REGRESSION: Nunally suggests that in multiple regression modeling, for each variable (X), there should be at least 10 counts, i.e. for Y = B0 = B1X1 + B2X2 --- then there should be 10 counts for X1 and 10 counts for X2 and 10 counts for Y or total 30 counts. See Nunally article.

REFERENCE: Some relevant article attached, others see citation below.

(1) Westland, J. Christopher (2010). "Lower bounds on sample size in structural equation modeling". Electron. Comm. Res. Appl. 9 (6): 476–487.

(2) Nunnally, J. C. (1967). "Psychometric Theory". McGraw-Hill, New York: 355.

(3) Yamane, Taro. 1967. Statistics: An Introductory Analysis, 2nd Ed., New York: Harper and Row.

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366.htm

Please refer to :

Testing for jumps in GARCH models, a robust approach

http://www.timberlake.co.uk/slaurent/pdf/BIP_GARCH_Main.pdf.

Minimum number: in the time series dimension T = 2, but then the size of the cross-section should tend to infinity. What is infinity? No less than 40-50.

With your sample size you have a problem because any size (in the time or firm dimension tend to infinity). Then, whether your size is enough is going to depend on the specification you have but even with a very parsimonious specification the sample size is very small.

Panel Data Econometrics by Badi Baltagi.

This attached file will help you a lot...

what should be the minimim length of daily observations for garch copula model?

60 or more than 60 are enough to run regression without concerning normality issue

I have data of 27 districts (cross sections) and daily time series of 15 years.

My question is: Can I apply Panel Data estimation techniques for estimation of model?

as much as will be better

It depends on how many variables you have.

5 years ok,

9 firms ok,

At least you should use 3 variables for each firm.