Results 1 to 5 of 5

Math Help - Regression: statistically significant difference between transformed coefficients

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    40

    Regression: statistically significant difference between transformed coefficients

    Hi All,


    I know it is straightforward enough to test for a statistically significant difference between two coefficients... but I have kind of a funny case that has gotten me rather confused (and a push in the right direction would be great). I am running 4 regressions. Let's say they are


    y = a1 + a2x1 + additional things
    y = b1 + b2x1 + additional things
    y = c1 + c2x1 + additional things
    y = d1 + d2x1 + additional things


    Where "additional things" is different between each regression. I am interested in the value F1 = exp(a2)/exp(b2) and F2 = exp(c2)/exp(d2). I want to test whether or not there is a statistically significant difference between F1 and F2, but this has thrown me for quite a loop because I am using both ratios and exponentials. Can anybody give me any insight in this respect? I would really appreciate it!


    Nick
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,165
    Thanks
    762

    Re: Regression: statistically significant difference between transformed coefficients

    Hey salohcin.

    Since exp(a-b)=exp(a)/exp(b) you will need the distribution of a-b and then the distribution of exp(a-b).

    If a2, b2, c2, and d2 are approximately normal then exp(a-b) will be approximately log-normal. If you have covariance between parameters then the covariance matrix of variables (i.e. covariance of each pair of variables) needs to be taken into account. You can estimate co-variance of two variables using standard techniques.

    Note that if the estimators you are using are maximum likelihood ones, then you can use results regarding estimators of functions of parameters as well as asymptotic results regarding normal approximations.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    40

    Re: Regression: statistically significant difference between transformed coefficients

    Hi Chrio,

    Thanks so much for your response. This is very helpful. I was hoping to ask a quick follow up if that is okay. I was realizing today I could do something much simpler. I could just check whether (a2-b2) - (c2-d2) is statistically different than zero using a plain old t-test. If this is not statistically different than zero, the exp(a2-b2) - exp(c2-d2) necessarily would not be statistically different than zero. Does this seem reasonable to you or am I fundamentally missing something? Thanks so much,

    Nick
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,165
    Thanks
    762

    Re: Regression: statistically significant difference between transformed coefficients

    You can do that, but you need to take into account the nature of the exponential function.

    Because of the non-linear nature of the exponential function, testing the non-exponential hypothesis will not capture the non-linearity which means that the variance will either be too small or too large depending on the nature of the transformation.

    I'd strongly recommend to look at the test statistic for your original hypothesis involving the exponential form if that is what you need to test.

    As an example, if our linear confidence interval was [1,2] then the exponentiated interval would be [2.71828,7.38906] which is a lot wider than the normal linear interval.

    Are you aware of how to get a test statistic for a function of a parameter given that the estimator is a maximum likelihood estimator (MLE)?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2010
    Posts
    40

    Re: Regression: statistically significant difference between transformed coefficients

    Thanks so much for your response! I am estimating all of this using OLS actually. I will go about this comparing my original specification (in exponentials). Thanks again
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 17th 2013, 04:24 PM
  2. Calculating Variances and Regression Coefficients
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: February 1st 2013, 10:01 PM
  3. Replies: 1
    Last Post: January 24th 2013, 12:26 AM
  4. Regression: Transformed variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: June 2nd 2011, 06:47 PM
  5. Replies: 1
    Last Post: March 14th 2010, 04:57 AM

Search Tags


/mathhelpforum @mathhelpforum