Joint hypothesis tests

When running regressions,

$$Y_t=\alpha+\beta_{9}x_{9,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (1)$$

$$Y_t=\alpha+\beta_1x_{1,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (2)$$

In equation (1), I test $$\beta_{9}=0$$, and find a p-value of 0.56

In equation (2), I test $$\beta_{1}=0$$, and find a p-value of 0.27

I then run a third regression

$$Y_t=\alpha+\beta_9x_{9,t-1}+\beta_1x_{1,t-1}+\beta_2x_{2,t-1}+\beta_3x_{3,t-1}+\beta_4x_{4,t-1}+\varepsilon_t (3)$$

In equation (3), I test, $$\beta_{9}=\beta_{1}=0$$, and get a p-value of 0.03.

Is there a reason why two variables may be insignificant individually but significant together. Does anyone have any useful books, references etc.

Thanks.

• I just want to be clear with terminology, is equation (1) actually: $Y_t = \alpha + (\beta_9 + \beta_2 + \beta_3 + \beta_4) x_{t-1} + \epsilon_t$ and if so how do you isolate the component of regression coefficient, particularly $\beta_9$. – Attack68 Jul 25 '18 at 16:00
• Sorry the x’s are different, I’ll change it, it should be x1 x2 etc – user22485 Jul 25 '18 at 18:55
• Ok, and then secondly what test are you performing or can you describe your procedure – Attack68 Jul 25 '18 at 19:16
• Lets call it OLS, then a test to see with two x's are significant in predicting the market, so b1=b2=0 – user22485 Jul 26 '18 at 10:21