The F-statistic in a multiple linear regression weighs the reduction in the residual some of squares (RSS) that is achieved by adding a number of covariates into the regression against the additional parameters needed to estimate their effects (the overall F-statistic compares the full model against the null model including only an intercept and no covariates) . If the overall reduction in RSS is moderate and the number of parameters large, the F-statistic may not be significant even if one or more of the betas are. With a single variable, the t-test of the hypothesis that beta=0 and the F-test for the simple linear regression are equivalent and a significant beta implies a significant F-statistic. The more "insignificant" or random explanatory variables you add to the model (i.e. the larger p, the number of parameters, becomes) , the less likely it is that the reduction in RSS achieved by the one significant variable justifies the addition of many (p) variables.

For example, if the null model (intercept only) has RSSnull=20 and a simple linear regression with variable x1 has RSS=15, the t-statistic becomes:

F = (RSS_null - RSS) / (p-1) / (RSS / (n-p))

With p = 2 and, for example, n=20:

F = 6

which is significant with a p-value of 0.0248

If adding a single random variable x2 to the model does not result in an appreciable reduction in RSS (say RSS=14.5), the overall model with p=3 is no longer significant:

F = 3.22

p = 0.0650

Thus the addition of random explanatory variables essentially "dilutes" the significant effect of a single important variable to the point that the overall model is no longer signficant.

Significant regression betas imply significant impact of any exogenous variable onto endogenous variable. But, that does not necessarily warrant the model fitness. F-statistic implies the model fitness. Sometimes, we consider a model consisting of exogenous variable/s, where the exogenous variable/s may have significant impact onto the endogenous variable (assuming no simultaneity), but the model is not a fit one, either because of want of parsimony or because of less important variable/s. In such cases, the individual exogenous variables will have significant impact onto the endogenous variable but the model may not be a fit one, leading to insignificant F-statistic.

thanks a lot Tapas for the opinion shared. Your point is well noted and very much appreciated

David,

What kind of analytic model are you referring to? More information would help us answer.

Pat

Patrick dear, I am referring to any simple multivariate linear regression model

for example, estimating the relationship

(wage)i = a + b(Male)i + c(educ)i + d(exper)i + ei

Thanks, David,

And b, c, and d are significant? What is the F testing? b, c, and d taken together?

Pat

Yes please

the F-test on the R2 is testing the null hypothesis that the 'true' R2 is zero. Because you can't reject that with 95% confidence, you should not interpret any parameter in your regression. Obtain more and better predictors, or change your dependent variable

The F-statistic in a multiple linear regression weighs the reduction in the residual some of squares (RSS) that is achieved by adding a number of covariates into the regression against the additional parameters needed to estimate their effects (the overall F-statistic compares the full model against the null model including only an intercept and no covariates) . If the overall reduction in RSS is moderate and the number of parameters large, the F-statistic may not be significant even if one or more of the betas are. With a single variable, the t-test of the hypothesis that beta=0 and the F-test for the simple linear regression are equivalent and a significant beta implies a significant F-statistic. The more "insignificant" or random explanatory variables you add to the model (i.e. the larger p, the number of parameters, becomes) , the less likely it is that the reduction in RSS achieved by the one significant variable justifies the addition of many (p) variables.

For example, if the null model (intercept only) has RSSnull=20 and a simple linear regression with variable x1 has RSS=15, the t-statistic becomes:

F = (RSS_null - RSS) / (p-1) / (RSS / (n-p))

With p = 2 and, for example, n=20:

F = 6

which is significant with a p-value of 0.0248

If adding a single random variable x2 to the model does not result in an appreciable reduction in RSS (say RSS=14.5), the overall model with p=3 is no longer significant:

F = 3.22

p = 0.0650

Thus the addition of random explanatory variables essentially "dilutes" the significant effect of a single important variable to the point that the overall model is no longer signficant.

May be you are facing problem of multi-colinearity. Plz check the pairwise correlation among your regressors. Among

Dear David:

I presume that you are referring to a model like:

Y= a + b.X1 + c. X2 +d. X3 + u

As you know, the fitted regression will give you a R-square and an adjusted R-square values. This is the reflection of "goodness of fit", which implies to what extent the variation of your right-hand side parameter estimates (say, a^ + b^.X1 +c^. X2 + d^X3) explains the variation of your left-hand side dependent variable, Y.

Now, the F-statistic of a regression essentially tests the null hypothesis: H0: R-2 = 0 against the alternative hypothesis H1: R-2 > 0. If, at all, you get any of the beta coefficients to be significant, there is no chance that your null hypothesis (as given) will be accepted, as at least some part of the regression is being explained. What you have suggested here seems somewhat like this... please remember that being an applied econometrician, who has been doing these exercises for the last 15 years, such a problem to me implies that there is something seriously wrong with your estimates (or interpretation).

However, the opposite is quite a possibility: that is none of the beta coefficients are significant, but your R-square values will be significant and high. This is a clear indication of multi-collinearity between the explanatory variables. The matrix of (X-transpose. X) will be too high, thereby making its inverse too low, because of the dependent column vectors (linear functions of each other), thereby resulting in statistically non-significant coefficients. The alternative is to drop variables under such circumstances or to run a step-wise regression so as to accommodate the best results.

Usually you should have the opposite case, i.e. significant F-statistic but insignificant betas because of multicollinearity. In your application, the problem is driven by the low R2 in the regression. Although you have found some effects, the overall modell is still rather poor, due to the huge heterogeneity of the individuals in your sample. Try to include more regressors or increase the sample size. With a high number of obs, the F-statistic will become significant even when the R2 is low.

Danke! Danke! Danke!.....Thanks a lot Koen, Franz, Muhammad, Nilanjan and Christian for the impressive suggestions made. Your respective suggestions are well noted and very much appreciated

There are two possible explanations ("being a statistician means never having to say you are certain")

The problem of multiple comparisons is at the bottom of this. On the one hand, the F-test is an overall (joint) test for all the regression coefficients, and if you are testing at a 5% level of significance, it means there is a 5% chance of getting a false positive result. On the other hand, if you do a t-test for each individual regression coefficient, each at a 5% level of significance, then there is an increased probability of getting one or more false positive results. This means that it is easier to get significant results if you do several (t-)tests than if you do only one (F-)test, given that all the tests are done at the same level of significance.

The other explanation is more mathematical. I'll explain in terms of two independent variables X1 and X2 with response variable Y. Suppose the model is Y=b0+b1X1+b2X2. The t-tests for b1 and b2 are t1 and t2 and the F-test for b1 and b2 jointly is F. If your design is orthogonal, (X1 and X2 uncorrelated) then

F=0.5(t1**2+t2**2).

However, if X1 and X2 are correlated, then

F=0.5(t1**2+t2**2+ 2*covariance term)

This covariance term could be positive or negative, depending on the signs of b1, b2 and the correlation coefficient between X1 and X2. If the covariance term is negative, then it means that F could be much smaller than either t1**2 and t2**2 in which case b1 and b2 could be significant individually but jointly not significant.

thanks a lot Francois for the wonderful point made...your suggestion is well noted and much appreciated

I am with Christian as a practical matter it will be the t-stats that are insignificant and the overall F-stat is significant. If you have an actual example where this has happened, it would be interesting to see but if all the coefficients are significant (especially if you control for the level of significance so it is consistent between the number of coefficients and the overall F-test) I can't imagine ever getting an insignificant f-stat.

Oh ok, thanks a lot John for the opinion shared. Your suggestion is well noted and much appreciated