What is the null hypothesis for the individual p-values in multiple regression?












3














I have a linear regression model for a dependent variable Y based on two independent variables, X1 and X2, so I have a general form of a regression equation Y = A + B1*X1 + B2*X2 + $epsilon$,
where A is the intercept, e is the error term, and B1 and B2 are the respective coefficients of X1 and X2. I perform a multiple regression with software (statsmodel in Python) and I get coefficients for the model: A = a, B1 = b1, B2 = b2. The model also gives me P values for each coefficient: Pa, P1, and P2. My question is: What is the null hypothesis for those individual P values? I know that the null hypothesis for variable X1 means that the model has a 0 coefficient for B1, but what about the other variables? In other words, If the null hypothesis is Y = A + 0*X1 + B2 * X2, what are the values of A and B2 for the null hypothesis from which the P-value for B1 is derived?






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  • 2




    Your model is missing an error term.
    – Andreas Dzemski
    2 hours ago
















3














I have a linear regression model for a dependent variable Y based on two independent variables, X1 and X2, so I have a general form of a regression equation Y = A + B1*X1 + B2*X2 + $epsilon$,
where A is the intercept, e is the error term, and B1 and B2 are the respective coefficients of X1 and X2. I perform a multiple regression with software (statsmodel in Python) and I get coefficients for the model: A = a, B1 = b1, B2 = b2. The model also gives me P values for each coefficient: Pa, P1, and P2. My question is: What is the null hypothesis for those individual P values? I know that the null hypothesis for variable X1 means that the model has a 0 coefficient for B1, but what about the other variables? In other words, If the null hypothesis is Y = A + 0*X1 + B2 * X2, what are the values of A and B2 for the null hypothesis from which the P-value for B1 is derived?






regression p-value







share|cite|improve this question









New contributor




tmldwn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 2




    Your model is missing an error term.
    – Andreas Dzemski
    2 hours ago














3












3








3


1





I have a linear regression model for a dependent variable Y based on two independent variables, X1 and X2, so I have a general form of a regression equation Y = A + B1*X1 + B2*X2 + $epsilon$,
where A is the intercept, e is the error term, and B1 and B2 are the respective coefficients of X1 and X2. I perform a multiple regression with software (statsmodel in Python) and I get coefficients for the model: A = a, B1 = b1, B2 = b2. The model also gives me P values for each coefficient: Pa, P1, and P2. My question is: What is the null hypothesis for those individual P values? I know that the null hypothesis for variable X1 means that the model has a 0 coefficient for B1, but what about the other variables? In other words, If the null hypothesis is Y = A + 0*X1 + B2 * X2, what are the values of A and B2 for the null hypothesis from which the P-value for B1 is derived?






regression p-value







share|cite|improve this question









New contributor




tmldwn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I have a linear regression model for a dependent variable Y based on two independent variables, X1 and X2, so I have a general form of a regression equation Y = A + B1*X1 + B2*X2 + $epsilon$,
where A is the intercept, e is the error term, and B1 and B2 are the respective coefficients of X1 and X2. I perform a multiple regression with software (statsmodel in Python) and I get coefficients for the model: A = a, B1 = b1, B2 = b2. The model also gives me P values for each coefficient: Pa, P1, and P2. My question is: What is the null hypothesis for those individual P values? I know that the null hypothesis for variable X1 means that the model has a 0 coefficient for B1, but what about the other variables? In other words, If the null hypothesis is Y = A + 0*X1 + B2 * X2, what are the values of A and B2 for the null hypothesis from which the P-value for B1 is derived?






regression p-value




regression p-value






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tmldwn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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tmldwn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









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edited 1 hour ago





















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asked 2 hours ago









tmldwn

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  • 2




    Your model is missing an error term.
    – Andreas Dzemski
    2 hours ago














  • 2




    Your model is missing an error term.
    – Andreas Dzemski
    2 hours ago








2




2




Your model is missing an error term.
– Andreas Dzemski
2 hours ago




Your model is missing an error term.
– Andreas Dzemski
2 hours ago










2 Answers
2






active

oldest

votes


















3














The null hypothesis is
$$
H_0: B1 = 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R},
$$
which basically means that the null hypothesis does not restrict B2 and A.
The alternative hypothesis is
$$
H_1: B1 neq 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R}.
$$
In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.



In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H_0$.



In my answer, I have not addressed the distribution of $epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like
$$
E[epsilon mid X1, X2] = 0
$$
then a similar conclusion holds asymptotically (under regularity assumptions).






share|cite|improve this answer























  • But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
    – tmldwn
    1 hour ago












  • A composite null hypothesis is a whole set of possible probability measures.
    – Andreas Dzemski
    1 hour ago










  • I have edited my answer to emphasize this point.
    – Andreas Dzemski
    43 mins ago



















0














You can make the same assupmtions for the other variables as the X1. The ANOVA table of the regression gives specific information about each variable significance and the overall significance as well.As far as regression analysis is concerned, the acceptance of null hypothesis implies that the coefficient of the variable is zero, given a certain level of significance.



If you want to acquire a more intuitive aspect of the issue, you can study more about Hypothesis testing.






share|cite|improve this answer





















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    2 Answers
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    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3














    The null hypothesis is
    $$
    H_0: B1 = 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R},
    $$
    which basically means that the null hypothesis does not restrict B2 and A.
    The alternative hypothesis is
    $$
    H_1: B1 neq 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R}.
    $$
    In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.



    In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H_0$.



    In my answer, I have not addressed the distribution of $epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like
    $$
    E[epsilon mid X1, X2] = 0
    $$
    then a similar conclusion holds asymptotically (under regularity assumptions).






    share|cite|improve this answer























    • But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
      – tmldwn
      1 hour ago












    • A composite null hypothesis is a whole set of possible probability measures.
      – Andreas Dzemski
      1 hour ago










    • I have edited my answer to emphasize this point.
      – Andreas Dzemski
      43 mins ago
















    3














    The null hypothesis is
    $$
    H_0: B1 = 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R},
    $$
    which basically means that the null hypothesis does not restrict B2 and A.
    The alternative hypothesis is
    $$
    H_1: B1 neq 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R}.
    $$
    In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.



    In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H_0$.



    In my answer, I have not addressed the distribution of $epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like
    $$
    E[epsilon mid X1, X2] = 0
    $$
    then a similar conclusion holds asymptotically (under regularity assumptions).






    share|cite|improve this answer























    • But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
      – tmldwn
      1 hour ago












    • A composite null hypothesis is a whole set of possible probability measures.
      – Andreas Dzemski
      1 hour ago










    • I have edited my answer to emphasize this point.
      – Andreas Dzemski
      43 mins ago














    3












    3








    3






    The null hypothesis is
    $$
    H_0: B1 = 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R},
    $$
    which basically means that the null hypothesis does not restrict B2 and A.
    The alternative hypothesis is
    $$
    H_1: B1 neq 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R}.
    $$
    In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.



    In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H_0$.



    In my answer, I have not addressed the distribution of $epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like
    $$
    E[epsilon mid X1, X2] = 0
    $$
    then a similar conclusion holds asymptotically (under regularity assumptions).






    share|cite|improve this answer














    The null hypothesis is
    $$
    H_0: B1 = 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R},
    $$
    which basically means that the null hypothesis does not restrict B2 and A.
    The alternative hypothesis is
    $$
    H_1: B1 neq 0 : text{and} : B2 in mathbb{R} : text{and} : A in mathbb{R}.
    $$
    In a way, the null hypothesis in the multiple regression model is a composite hypothesis. It is "fortunate" that we can construct a pivotal test statistic that does not depend on the true value of B2 and A, so that we do not suffer a penalty from testing a composite null hypothesis.



    In other words, there are a lot of different distributions of $(Y, X1, X2)$ that are compatible with the null hypothesis $H_0$. However, all of these distributions lead to the same behavior of the the test statistic that is used to test $H_0$.



    In my answer, I have not addressed the distribution of $epsilon$ and implicitly assumed that it is an independent centered normal random variable. If we only assume something like
    $$
    E[epsilon mid X1, X2] = 0
    $$
    then a similar conclusion holds asymptotically (under regularity assumptions).







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 44 mins ago

























    answered 2 hours ago









    Andreas Dzemski

    1915




    1915












    • But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
      – tmldwn
      1 hour ago












    • A composite null hypothesis is a whole set of possible probability measures.
      – Andreas Dzemski
      1 hour ago










    • I have edited my answer to emphasize this point.
      – Andreas Dzemski
      43 mins ago


















    • But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
      – tmldwn
      1 hour ago












    • A composite null hypothesis is a whole set of possible probability measures.
      – Andreas Dzemski
      1 hour ago










    • I have edited my answer to emphasize this point.
      – Andreas Dzemski
      43 mins ago
















    But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
    – tmldwn
    1 hour ago






    But as I understand it, doesn't the null hypothesis have to be a probability distribution? If I have specific values for the coefficients, I can generate a probability distribution by adding noise (epsilon) to the regression equation. But if I don't have specific values for coefficients, how would I generate the null probability distribution?
    – tmldwn
    1 hour ago














    A composite null hypothesis is a whole set of possible probability measures.
    – Andreas Dzemski
    1 hour ago




    A composite null hypothesis is a whole set of possible probability measures.
    – Andreas Dzemski
    1 hour ago












    I have edited my answer to emphasize this point.
    – Andreas Dzemski
    43 mins ago




    I have edited my answer to emphasize this point.
    – Andreas Dzemski
    43 mins ago













    0














    You can make the same assupmtions for the other variables as the X1. The ANOVA table of the regression gives specific information about each variable significance and the overall significance as well.As far as regression analysis is concerned, the acceptance of null hypothesis implies that the coefficient of the variable is zero, given a certain level of significance.



    If you want to acquire a more intuitive aspect of the issue, you can study more about Hypothesis testing.






    share|cite|improve this answer


























      0














      You can make the same assupmtions for the other variables as the X1. The ANOVA table of the regression gives specific information about each variable significance and the overall significance as well.As far as regression analysis is concerned, the acceptance of null hypothesis implies that the coefficient of the variable is zero, given a certain level of significance.



      If you want to acquire a more intuitive aspect of the issue, you can study more about Hypothesis testing.






      share|cite|improve this answer
























        0












        0








        0






        You can make the same assupmtions for the other variables as the X1. The ANOVA table of the regression gives specific information about each variable significance and the overall significance as well.As far as regression analysis is concerned, the acceptance of null hypothesis implies that the coefficient of the variable is zero, given a certain level of significance.



        If you want to acquire a more intuitive aspect of the issue, you can study more about Hypothesis testing.






        share|cite|improve this answer












        You can make the same assupmtions for the other variables as the X1. The ANOVA table of the regression gives specific information about each variable significance and the overall significance as well.As far as regression analysis is concerned, the acceptance of null hypothesis implies that the coefficient of the variable is zero, given a certain level of significance.



        If you want to acquire a more intuitive aspect of the issue, you can study more about Hypothesis testing.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        Logicseeker

        183




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