from ANOVA table point of view,

The sum of square due to estimation (ESS) is [S]y(i)_hat^2=b2_hat^2 * [S]x(i)^2, where [S] is the usual sigma, summation sign. the df for this item is one. This is understood as follows. we assume x(i) are known (pre-dertermined or conditioning variables), so df has nothing to do with this (some sort of). we say that b2_hat has one df because we have shown that b2 has taken one df. so we think b2_hat "holds" one df. since ESS is a function of b2_hat only, so ESS has one df.

Note that if b2_hat holds one df, so is b2_hat^2, or f(b2_hat), where f(.) is a well behaved function.

Let us come to Residual sum f sqaures (RSS). RSS=[S]u(i)_hat^2. we initially have n obs. but taking mean of Y(i) takes one df away (such that we have b1_hat, the intercept), estimating b2 also reduces one df. now only n-2 df are left for RSS.

The Total sum of sqaures, TSS=[S]y(i)^2 has n-1 df since we started from Y(i), took mean of Y(i), we lost one df when we calculate Y_bar. So n-1 df for TSS.

any standard text should have something akin to the above. Then this makes life simple when it comes to F test. the numerater df is 1 and the df for the denominator is n-2, such that the F(1),(n-2) is in order.

Some other intuition provoking comments

(1) degrees of freedom are the equivalent of currency in statistics - you earn a degree of freedom for every data point you collect, and you spend a degree of freedom for each parameter you estimate. Since you ususally need to spend 1 just to calcultae the mean, you then are left with n-1 (total data points "n" - 1 spent on calculating the mean).

(2)

Another way of thinking about degrees of freedom is to image that you ask 100 people to pick a number between -infinity and infinity so that the mean of all the numbers is exactly zero. 99 of the people can pick any number they want, but the 100th person HAS to pick a specific number. He is not "free" to pick what he wants. Thus the degrees of freedom to determine the mean for 100 samples is n-1 or 99.

(3)

Perhaps a physical depiction will help clarify this concept. Take a piece of rope and hold the two ends. Now twirl it (like you did in high/junior school gym) so that it starts moving in the middle (like a skipping rope). You had two nodes and when you twirl it you find that the rope is able to move at one point only. It has the freedom to move at one point only! It has two nodes and one degree of freedom. Now ask someone to hold it at the center point thus creating another node; you now have 3 nodes. If you try twirling it now, you find that the rope can be twirled at 2 points. Three nodes, two degrees of freedom!!

This might be a real basic explanation and mathematicians might disagree with me, but it beats any other way of explaining Degrees of Freedom.

(6)

You could think of Degrees of Freedom as the amount of information available for the estimation of variability. For example, the degrees of freedom to estimate variability from one observation would be zero, i.e., it is impossible to estimate variability. But for each observation after the first, you get one addtional degree of freedom to estimate variance. Hope this helps.

(7)

The best explanation I've heard for understanding degrees of freedom in various statistical calculations is as follows:

Degrees of freedom of n - 1 is required when taking a sample from a population because when taking the limited size sample, you have only a very slight chance of picking the extreme data values of the population. In order to accomodate for this, you subtract the value of 1 (n - 1) in order to inflate the std deviation and make the standard deviation calulation more adequately represent the parent population.

Notice that in the calculation of the population std deviation, the sum of squares is divided by the value N and not n - 1. This is because all values in the poplulation (including extreme values) are taken into account.

(9)

My BB coach taught the same lucid way with a different example. It is also important to note that the neccesity to subtract 1 from n becomes negligible as n increases in size where n > 30.

(zhaosweden: of course, when sample size increase, loss in df in not important, but when sample size is small, we prefer simple model, such that we can estimates things more accurately.)

[此贴子已经被作者于2006-3-12 3:30:02编辑过]

Estimates of parameters can be based upon different amounts of information. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df). In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. For example, if the variance, σ² , is to be estimated from a random sample of N independent scores, then the degrees of freedom is equal to the number of independent scores (N) minus the number of parameters estimated as intermediate steps (one, μ is estimated by M) and is therefore equal to N-1.

(c)

DEGREES OF FREEDOM EXPLANATION

Sampling and Statistics

When the population standard deviation ( ) is unknown and the sample size is less than 30 (n < 30), the distribution of the test statistic can not be guaranteed to be normal. In fact, the test statistic can be said to conform to what is called a t distribution.

The t distribution is similar to the standard normal distribution in that it is symmetrically distributed around a mean value. But where the t distribution varies from the standard normal, is that its standard deviation is determined by what is known as the number of degrees of freedom.

Degrees of freedom are calculated from the size of the sample. They are a measure of the amount of information from the sample data that has been used up. Every time a statistic is calculated from a sample, one degree of freedom is used up.

When you have a very large sample drawn from your data set, the difference between t values and Z values is miniscule. But as the size of your sample falls, the t distribution takes on a standard deviation increasingly greater than 1. In other words, this means that the t distribution when n < 30, is more spread out than the standard normal distribution.

OK, let me stop here. In above, I made my own comment and pasted some internet materials. I hopw when you are patient enough to go through them, you must have had a better understanding of "Degrees of freedom" such that you now have more degrees of freedom in Econometrics!

This concept can always be confusing to entry level students. There are also degrees of freedom concept in Physics and Chemistry, If you are interest, you can type such keywords in google.

Moreover, the zipped file is a Word DOC file of some website explanation, I think it is very nice.

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• Degrees of freedom1.doc

[此贴子已经被作者于2006-3-11 22:12:36编辑过]