Hello, I am estimating some non-linear models and I need to estimate the corresponding one-sided z-statistic for each regressor and a two sided z-statistic for the intercept. Could you please provide some guidance for their computation, as well as their p-values? Thank you, Karla

## 3 Answers

As an example, consider the non-linear regression results below, obtained using simulated data and the **Maximum Likelihood MT** application module:

As an example, consider the non-linear regression results below, obtained using simulated data and the **Maximum Likelihood MT** application module:

Log-likelihood -208.823 Number of cases 200 Covariance of the parameters computed by the following method: ML covariance matrix Parameters Estimates Std. err. Est./s.e. Prob. Gradient --------------------------------------------------------------------- b0[1,1] 0.5485 0.2488 2.205 0.0275 0.0000 b[1,1] 0.0693 0.0567 1.223 0.2214 0.0002

In this example, the probabilities listed in the table both correspond to two-sided tests of the hypothesis that b0[1,1]≠0 and that b[1,1]≠0. We can find both the two-side p-values in table and one-sided p-values using GAUSS procedure ** cdfN**. This procedure computes integral of Normal distribution: lower tail, or cdf for a given z-statistic. Hence, to compute the hypothesis test that a coefficient is different from a null hypothesis, h0, we first need to find the z-statistic:

//Null hypothesis h0=0; //Z_Stat = (estimate - h0)/se //Find intercept Z-statistic Z_b0 = (0.5485 - h0)/0.2488; //Find b1 Z-statistic Z_b1 = (0.0693 - h0)/0.0567;

Next, to find the probabilities corresponding to the one-sided test:

//Find one-sided p-value Z_b0 p_onesided_b0 = 1-cdFN(z_b0); //Find one-sided p-value Z_b1 p_onesided_b1 = 1-cdFN(z_b1);

Finally, to find the two-sided p-values (listed in the table), we multiply the one-sided by values by 2:

//Find two-sided p-value Z_b0 p_twosided_b0 = 2*p_onesided_b0; //Find two-sided p-value Z_b1 p_twosided_b1 = 2*p_onesided_b1;

Thank you very much for answering my question. Could you please also provide a way of computing the **one-sided critical values |z|** for significance levels 99%, 95% and 90%, assuming a sample size of 700 observations? Thanks again.

To find the one-sided critical value for a normally distributed variable, let's start from the equations in my original response for finding the p-value for an associated z-statistic:

```
//Find one-sided p-value Z_b0
p_onesided_b0 = 1-cdfn(z_b0);
```

If we rearrange this equation we get:

//Find two-sided p-value Z_b0 cdfn(z_b0) = 1 - p_onesided_b0;

For the 99%, 95%, and 90% significance levels, the associated probability values, α, are equal to 1%, 5%, and 10%. Note, additionally, that α is equivalent to the variable p_onesided_b0. So if we use 99% as an example:

//Find two-sided p-value Z_b0 cdfn(z_b0) = 1 - 0.01 = 0.99;

To solve this for the associated critical values we can use the GAUSS function cdFni. This function is the inverse of cdFN. To find the critical value for a 99% significance level:

//Find two-sided p-value Z_b0 z_b0 = cdfni(0.99);

We can also easily find this critical value using the GAUSS command line by typing cdFNi(0.99) and pressing enter:

>>cdfni(0.99) 2.3263479 >>cdfni(0.95) 1.6448536 >>cdfni(0.90) 1.2815516