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I understand how Wald testing is derived i.e.

(Sorry, not sure how to add formulae to these questions)

$$\dfrac{(RB-q)'(R(X'X)^{-1}R')(RB-q)}{s^{2}}$$

This will be distributed as F(M, N-K), where M is the number of restrictions and I am assuming that the coefficient vector B is normally distributed. R is the matrix of restrictions, and K the number of regressors.

In practice, I understand that instead of just estimating the unrestricted model and then calculating the above statistic, statistical packages estimate both the restricted and unrestricted models, and create an identically distributed statistic which is: $$ \frac{(SSR_r-SSR_{ur})/m}{(SSR_{ur})/(N-K)} $$ So my question is specifically, how is the restricted version of the model estimated when the restriction is that a ratio of two of the coefficients is equal to 1 (for example)?

Imagine the regression is $Y_t= C + B_1X_t + B_2Z_t + u_t$

The restriction to test is $B_1/B_2 = 1$

Would this need to be done via non-linear regression?

gung - Reinstate Monica
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Alexander Whyte
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  • Added one equation for you. If you have a look, you might see how to do it. It's not too difficult. (Also check it's right. :) – Jeremy Miles Jan 01 '17 at 03:04
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    That's right! Thank you - Yes, I'll take a look and see if I can update the other ones. – Alexander Whyte Jan 01 '17 at 03:07
  • Just translate it to the equivalent linear restriction that $B_1=B_2$. – Christoph Hanck Jan 01 '17 at 16:03
  • I considered that, but the problem is that I know that Wald tests are not invariant to parametarization, so stat programmes like Eviews will give you a different statistic if you specify the test in that form. So I am just wondering how it is that Eviews, or Stata etc provide the test statistics when you enter a wald test of the form I mentioned when the original regression has been run via OLS. – Alexander Whyte Jan 01 '17 at 16:35

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