What are the standard hypothesis tests in any type of regression?

In general, what the the standard hypothesis tests in all types of regression? We get some coefficients and then test the hypotheses that these coefficients are significant (i.e not equal to $0$)? For example, in logistic regression the p-value indicates that the regression coefficients are significant or not? Likewise with linear regression?

Question: Suppose coefficients $\beta_0$ and $\beta_1$ are significant in linear regression. Can we deduce anything about the significance of $\beta_0+\beta_1$? What about $\beta_0= \beta_1$?

• Indeed, softwares return the $p$-value for $H_0:\{\theta=0\}$ for each parameter $\theta$ of the model. – Stéphane Laurent Jan 18 '13 at 20:00
• @StéphaneLaurent: Is there a way to test for example $H_0: \theta_1+\theta_2 = 2$? – proton Jan 18 '13 at 20:02
• Is this homework? – Stephan Kolassa Jan 18 '13 at 20:14
• @proton For a Gaussian linear regression model it is easy to test about any linear combination of the parameters. – Stéphane Laurent Jan 18 '13 at 20:16
• @StéphaneLaurent: How about for other types of regression. – proton Jan 18 '13 at 20:21

You can create a contrast matrix $L$ and a vector of parameters $\beta$. Then test any kind of linear combination of the parameters using the null hypothesis, $H_0:L\beta=d$. You can use a Wald test or a Likelihood Ratio Test (LRT) for testing your $H_0$.
• What if you want to test $H_0: \frac{\beta_0}{\beta_1} = 1$ vs $H_a: \frac{\beta_0}{\beta1} \neq 1$? – proton Jan 21 '13 at 14:56
• It is all the same as $H_0:\beta_0-\beta_1=0$, isn't it? More over you could also test $H_0:\frac{\beta_0}{d\beta_1}=1$ by constructing a contrast matrix $L$ such that $H_0:\beta_0-d\beta_1=0$. – Blain Waan Jan 21 '13 at 18:36