Hi Adrian,

thanxx for replying, you know i used to restrict the number of explanatory variables, but the issues came up again(e,g with the signs or with p-value of the variables.)

but i will see the first one .

SAMPLE SIZE

If all available data available is 21 observation, assume that 21 observation is an initial sample and that the population is nonfinite, i.e. unknown. The question arises: What is the required minimum sample size? For nonfinite population, the minimum sample size formula is given by:

(1) n* = [Z^2(sigma^2)] / E^2

… where n* = minimum sample size, Z = unit normal distribution critical vale of 1.65 for two-tailed test at 0.05 error level; sigma^2 = assumed variance; and E = sampling error or SE where SE = 1/ sqrt (n).

Note that sigma may be determined indirectly through the t-equation and Z-equation in a 2-steps process.

(2) t = (x^ - mu) / S / sqrt n

… x^ = sample mean; mu = assumed population mean; S = sample standard deviation and n = sample size. From the t-equation, mu can be determine by allowing t = 1.64 for infinity degree of freedom (as a standard practice).

(3) mu = t(S / sqrt n) – x^

After mu is determine, substituted into the Z-equation. The Z-equation is given by:

(4) Z = (x^ - mu) / sigma / sqrt n

Now solve for sigma:

(6) sigma = [(x^ - mu) / Z] sqrt n

Use the value of sigma from equation (6) to put in place of sigma in equation (1), the minimum sample size is determined. Note that throughout this process, since only 21 observations are available, one may want to partition it into two samples: n1 = 10 and n2 = 11 and follow the above process and compare n* from both cases.

TEST OF HOMOGENEITY

If a model is to be constructed from the 21 observations, it is necessary to test whether these 21 elements are homogeneous. This can be accomplished by paired t-test or d-bar analysis. Remove one observation to make 10 pairs from n:{(x1, x2, …, x10), (y1, y2, …,y10)}. Now pair (x:y) to find the paired difference:

(x1 – y1) = d1

(x1 – y1) = d2

…

(x10 – y10) = d10

The mean is given by:

(7) d^ = (1/n)(sum d,)

and standard deviation:

(8) S(d) = sqrt ((1/n) sum (d – d^)^2)

The test statistic is given by:

(9) t(d) = d^ / S(d) sqrt (n)

Assume that the confidence interval is 0.95 and the degree of freedom is infinity, the critical value is t =1.64. If the t-critical score for the observation is less than 1.64, then the 20 observation is homogenous. Thus the 21 observations can fairly be assumed to be homogeneous. With this homogeneity proven, it is now ready to be tested for distribution type. Is it normally distributed?

ANDERSON-DARLING TEST

According to the Anderson-Darling, the minimum sample size is n >5 or at least 6 elements. The test is used to determine the characteristic of the data distribution. In case where the population mean and variance is unknown, the critical value is 0.787 for 0.95 confidence interval. The test statistics is given by:

A^2 = -n – S

… where A^2 = Anderson-Darling score; n = sample size; and S = sum ((2k -1)/n)[ln (F(Y)) + ln(1-F(Y^n+1-k))]

The advantage of this test allows the research to redefine the population to a smaller size. In this case, if the available sample is 21; it can be redefined as a “population” of 21 and a sample of 6 can be drawn in a re-sampling process akin to bootstrapping thus: n1 = 6 then replace (meaning read the sample statistics and throw back the sample into the set to maintain N = 21, then n2 =6 is taken, … ni = 6). The law of large number will manifest the true character of the “population” of 21.

Under the condition thus described “ N = 21;” however, the true N = true population size is not known. Is it necessary to know the true population size in order to construct an accurate model? No. Unknown population is the condition commonly found in real life situation. See minimum sample size for non finite population, supra.

CAUTION OF SMALL SAMPLE SIZE

Since the sample is small, one may miss the true characteristic of the population. Assume that the available sample is fixed at 21 elements and no more could be obtained, the model can still be constructed with this sample size. Use the Anderson-Darling test to determine the characteristic of the data distribution, i.e. whether it is normally distributed. If the sample is small, but the Anderson-Darling test confirms that it is normally distributed, confirm the distribution by replaced re-sampling from the set of 21 elements---meaning re-sample 6 elements at a time and run the test. If the slope sign (+ or -) is consistent, it means that the regression line points to the same direction in all subsamples; if the slope sign changes, it means that there is a curve in the data distribution (nonlinear). This slope sign test will allow the researcher to construct the model accordingly: (i) linear, (ii) exponential, (iii) logit, etc.

I have used one-delete jackknife re-sampling plan with random draw for deletion. Use random number generator for deletion selection. You can do this a reasonable number of times to obtain reasonable model parameter estimates.

Article On a solution for the high-dimensionality-small-sample-size ...

http://www.statisticssolutions.com/sample-size-formula/

https://measuringu.com/small-n/

Nagma Hbiesh me too i have a linear time series of 13 years, what did you do to solve the problem please i need your answer, because some say that i can not use ARDL since it requires at least 30 observations

I think the best way is to find a way of converting your , say annual series into quarterly or monthly data, that way you can easily increase your sample probably beyond 30.

Hello,

You may need to use non parametrical statistical tool for the 21 cases. Non parametrical tools are less powerful but can cater for situations when minimal sample size has not been met.

You might want to see the following:

Article Small sample sizes: A big data problem in high-dimensional d...

https://www.nature.com/articles/nrn3475

https://erj.ersjournals.com/content/erj/32/5/1141.full.pdf

Article How sample size influences research outcomes

https://www.statisticssolutions.com/what-to-do-when-your-sample-size-is-not-big-enough/

Actually while using econometrics use minimum of 800 plus data, then only it will show more accuracy results