# Comparing two models using Wald Test

I would like to use a Wald test to compare two models.

To give a basic example, let:

$Y_{t}=\alpha+{\phi_1x}_{t-1}+{\beta}_{1}x_{t-1}+{\beta}_{2}x_{t-1}+{\beta}_{3}x_{t-1}+\epsilon_t (A)$

$Y_{t}=\alpha+{\beta}_{1}x_{t-1}+{\beta}_{2}x_{t-1}+{\beta}_{3}x_{t-1}+\epsilon_t (B)$

I have completed a Wald test to test the joint significance of each of the dependent variables. Can anyone see anything statistically wrong with comparing the Wald stat from the two equations.

And apart from this, if the the p-value for model A is larger, does it rule out $\phi$ ?

• What is $\phi_{t-1}$? Is that data or are you estimating it? Where are your right hand side variables? (I assume the betas are coefficients on some data?) What is your precise question? – Matthew Gunn Mar 8 '18 at 17:55
• Hi thanks for the reply, treat phi as one of the betas, it’s just an extra lagged variable. – user22485 Mar 9 '18 at 10:55
• Your variables and subscripts do not make sense. For example, is model A actually $Y_{t}=\alpha+{\beta}_{1}x_{1,t-1}+{\beta}_{2}x_{2,t-1}+{\beta}_{3}x_{3,t-1}+ {\phi}x_{4,t-1}+\epsilon_t$? – Matthew Gunn Mar 9 '18 at 15:43
• Yes sorry Matthew, i was just trying to write something down quickly every parameter has a variable. – user22485 Mar 9 '18 at 15:47
• It sounds like you want to statistically test the difference between nested models. If you're adding one predictor, you can just do a t-test on the coefficient for that predictor. More generally, you can use an F-test to compare nested models (if difference is more than a single predictor). If models aren't nested, you can use AIC. – Matthew Gunn Mar 9 '18 at 18:22