You can do a very simple thing:  1) subtract the mean of age from your original variable ; 2) after take the square of this de-meaned age; 3) regress you dependent var on the demeaned age and its square.  Now, you have reduced collinearity problem, furthermore the interpretation of the age-coefficient remains the same.

You can compute the VIF (http://en.wikipedia.org/wiki/Variance_inflation_factor) and see how such a collinearity may influence your empiricial results. It can be implemented easily in stata

Of course, age and age squared are correlated. However, one is not a linear transformation of the other by definition. I think that this is what Trung means. I doubt very much that you will have an econometric problem here.

Another option beyond what Gabriele said, is to normalize the variable age and create an age squared term.   

Nguyen is right, in most studies involving "age", it is always a good idea to include the squared "age" variable in order to reduce the effect of age classification. the collinearity is present but, not damaging!

nope.. and a simple VIF test can be done to confirm it.

Another possibility is to run a regression between Age and Age2  to see that the R2 is close to zero. This indicates, as mentioned before by other colleagues, that collinearity is very low and can be overlooked from a econometric perspective.

Hold your horses here, Andrew and others. You cannot just willy nilly rescale nonlinear variables without econometric consequences. The issue here is what you are trying to accomplish by modeling with a squared term. For example, suppose that the dependent variable is property price or its log and suppose further that you believe that as properties age their prices decline at first but then rise later as buildings acquire value as antiques. Without rescaling the variables, the linear age term might be expected to be negative while the squared age term might be expected to be small and positive. Alternatively, if you rescale . . . but do not get the right point for the minimum of the age function (i.e., the average age may not be anywhere near the minimum). The results might be much worse: you might miss capturing your hypothesized relationship when it actually exists.

Calculate the covariance x and x 2, you will find zero covariance, as covariance is is a linear relationship. 

ISSUE: How to treat age?

DATA CLASSIFICATION: The question involves data classification. Generally, age, sex, education, etc. are part of the demographic information of the section. It is seldom that one uses these demographic data as explanatory factor, i.e. X = independent variable. In this case, "age" is used as part of Xi.

Recall that age data is generally collected as "age group." For example, 20 - 25; 30 - 45; 45 - 55; and above 55. (We do not sum all these "ages" and then take the mean because age is used for identification not as data subject to mathematical operation without first coding it). This type of data is called "nominal." Nominal data is used for purposes of identification. In this case, to identify age group, i.e. how many people could be identified in that age group. If age is used as part of the explanatory (independent variable), it must be coded first. The coding available for nominal data is 1 and 0. The score code of 1 is used for the category of interest and outside of that category is coded as 0. For this particular data set: (1, 0), the distribution follows discrete distribution:

(1)   lim1-inf (Sn - np) / sqrt(npq)

... where lim = limit from i = 1 to infinity; Sn = summation of all x1, x2, ... xn which is the sum of all 1 because 0 is zero value. the definition of p and q is the Laplacian rule of succession:

(2)   p = (s + 1) / (n + 2)

... where s = score of 1 or targeted age group; n = sample size or total observation. the value for q is simply 1 - p.

Now... assume that your dependent variable (Y) is quantitative, in the data classification matrix, this falls in (YQ;XN) requiring point biserial correlation test.

SQUARING AGE: As explained above, age is scored as 1; thus, 12 = 1. FROM the brief explanation in the question, it appears you are trying to run collinearity test among Xi. This can still be accomplished. Assume now that your other Xi is a quantitative measurement, say, you want to run collinearity test between age and X1. Point biserial correlation follows:

(3)   rpb = [(M1 - M0) / Sn]A

... where A = sqrt[(n1(n0)) / n2]; M1 = mean for group of data with score of 1 (these are the target age group); M0 = mean of groups with score 0; and S = standard deviation. As for n1 and n2, similar definition applies.

The value of r ranges from -1 to =1. The test statistic to accompany rpb is given by:

(4)   tpb = rpb[sqrt(n1 + n2 -2) / 1 - r2pb]

Set you confidence interval and compared the observed tpb to that in the Z-table. you should be able to conclude whether age (a specific age group) is correlated to you other explanatory variable.

I would:

1 - check the correlation and variance inflation factor between the two. Depending on the range you are looking over, you may not have a huge multicollinearity problem.

2. - If you do, then center the x variables before squaring them, as described by Gabriele.

Frances claims wright. High correlation between independent variables doesn't cause collinearity. You had better checked VIF at first. See " regression application by John Sall (SAS Techinical report" or my articles about CPD data. There are three strong multi-collinearities. On the other hand, Swiss bank note data has six measurements. Therefore, there are high correlation, but all VIFs are small.  

Note that you probably estimate a linear regression model. Therefore, you should  be aware of linear dependencies. The correlation between age and (age)**2 is 0, since there is no linear relationship. So no problem with the quadratic dependency. 

In my opinion you will always find correlation between age and agesquare. They co-move, when age is increasing also its square is increasing,  the covariance will not never be zero as written above and therefore also the correlation will be not zero. See for instance the attacched file. You have for sure a curvilnear relation between the variable that I call age and its square, however the straight line approximates very well the relation and you have an R square equal to 0.93. 

This thread has really gotten off track. Attached are some "data" that I just created. You will find an age column and an age squared column. You will also find columns for these magnitudes that have been "normalized" with respect to the average age only. Below these columns are two correlation coefficients. Note that the raw data have a high correlation  whereas the "normalized" data have a low correlation. This first table is populated by randomly drawing the age data. On the right of this table is another table that has been frozen (copied and pasted with only values and formats) with unchanging magnitudes. For those who are not visualizing what I have done here, I would recommend graphing the age and age squared data. The correlation depends upon where the observations are on these two functions.

I attach the Excel file that include four independent variables and two object variables.

I analyse two regressions such as :y1/y2=a1*age+a2*age^2+a3 and y1/y2=a1*(age-m)+a2*(age-m)^2+a3. VIF=17.8 and 1. So, there are not multicollinearity in these regression models.

Multicollinearity is NOT a problem in your case. Multicollinearity has to be checked and problems have to be solved when you want to estimate the independent effect of two variables which happen to be correlated by chance. This is NOT your problem with your age and agesquared variables since you should never be interested in evaluating the effect of changing age without changing agesquared. So do not care about multicollinearity between one variable and a second variable which is a deterministic non linear function of the first one. Except in the case of perfect multocillinearity, which would be the case if you had only two different values for your age variable.

you said col-linearity; you model is not lineal. period.

While perfect collinearity (a regressor which is a linear combination of other regressors) is troublesome for OLS, imperfect collinearity is not at all a failure of assumptions underlying your estimator and you should not correct your model for it. Collinearity is a feature of your data: could you increase your sample size, it would disappear. If you have solid theoretical ground for including age squared, do it. In case you cannot check the separate effect of age and age squared, you can nonetheless test their joint effect.

The classical reference is: "A Course in Econometrics" by Arthur S. Goldberger, chapter 23. Let me also add a reference to a very short article which goes somewhat deeper in the subject: http://www.stata-journal.com/article.html?article=st0285

Age and Age2 had better be correlated. Adding more data will not change the effect of collinearity. 

If you consider a designed experiment, the software you use will code your variables so that they range from (-1 to 1). It will determine the VIF of your data when it is coded. Then expand the coefficients so they reflect the real range of values. The VIF for most terms in a designed experiment are 1.00. Even the squared terms and 2-way, 3-way, 4+-way interactions. Cubic terms get a bit messy, A2B and other terms like are not. 

If you apply this idea to observational data, the VIF among your main effects will not change. The VIF among your interactions will drop into a more acceptable range. This will let you find more interactions and quadratic terms that might be important, without this issue of large collinearity. It will then be up to you to figure out which terms really are important and which are not. 

High correlation does not mean the collinrarity. VIF detects collinearity. Swiss bank note data has 6 variables and high correlation. But there is no collinearity. See my paper about the CPD data and Swiss bank note data after 2014.

Hi Peter F. Colwell,

I also had a similar problem and followed your advise and worked pretty well. Is there any source I can cite to justify the normalization of the age variables?

See this post from Paul Allison on collinearity caused by inclusion of powers (#2):

https://statisticalhorizons.com/multicollinearity

Copying his point verbatim here:

If you specify a regression model with both x and x2, there’s a good chance that those two variables will be highly correlated. Similarly, if your model has x, z, and xz, both x and z are likely to be highly correlated with their product. This is not something to be concerned about, however, because the p-value for xz is not affected by the multicollinearity.  This is easily demonstrated: you can greatly reduce the correlations by “centering” the variables (i.e., subtracting their means) before creating the powers or the products. But the p-value for x2 or for xz will be exactly the same, regardless of whether or not you center. And all the results for the other variables (including the R2 but not including the lower-order terms) will be the same in either case. So the multicollinearity has no adverse consequences.

Your model is not linear. So there is no problem. If you modify your variables, you will face a difficulty in interpreting your estimated parameters which would confuse with your initial estimation goal.