Issues in testing a simple linear relationship. Collinearity? Misspecification? Any other insight?

Question

I have a theoretical model saying that Y should be equal to:

Y = X + c * (W - X) + (Z1 - Z2),

where c is a given constant.

Here, it may be important to say that X is measured with error.

Someone proposes to test that assumption by estimating a linear model:

Y = b0 + b1 * X + b2 * c * (W - X) + b3 * (Z1 - Z2) + u,

and testing that b0 = 0, toghether with b1 = b2 = b3 = 1.

In my concrete estimation, I found that the estimators of b0 and b1 are highly correlated, about -0.80.

When I include the intercept, the estimate of b1 is about 0.75, significantly different from 1.

When I do not include the intercept, the estimate of b1 is very close to 1, 1 being in the corresponding confidence interval at 95%, and thus validating part of the story.

Regardless of the intercept, all the other coefficients are never close to 1, so that I am constantly rejecting the null hypothesis of b0 = 0, b1 = b2 = b3 = 1, altogether.

If I reformulate:

Y2 = Y - c * W - (Z1 - Z2),

such that:

Y2 = (1 - c) * X,

when I estimate Y2 = b1 * (1 - c) * X without intercept gives me an estimate of b1 very close to 1, with 1 being in the interval.

Any guess of why?

I am tempted to present results from the second model, which estimate only 1 coefficient, and the results are in line with expectations.

However, I am not really seeing the reason why the results from the two models differ so much, and especially why the estimates from the first model are so far from the theory.

Any insight? :-)

Thank you very much!

Answer 1

In the first regression, the third term $c*(W - X)$ has $X$ as a component. This will make the second and the third term correlated. Consequently, the OLS estimates of the coefficients will be inaccurate (i.e. large standard errors).

The regression can be rewritten as $$ Y = \beta_0 + \beta_1X^{*}+\beta_2W^{*}+\beta_3Z^{*} + \epsilon$$ where $X^{*} = (1-c)*X$, $W^{*} = c*W$, $Z^{*} = Z_1 - Z_2$. This formulation avoids the correlation issue. The validity of the theoretical relationship can then be tested as $H_o: \beta_1 = \beta_2 = \beta_3 = 1$.

But the above formulation is workable if there is no measurement error. Since you mentioned $X$ is measured with errors, what is actually observable is $\tilde{X}$, where $\tilde{X} = X + \eta$ and $\eta \sim N(0, \sigma_x)$. The errors-in-variable regression is then

$$ Y = \beta_0 + \beta_1X^{**}+\beta_2W^{*}+\beta_3Z^{*} + \gamma$$ where $ X^{**} = (1-c)*\tilde{X}$ and $\gamma = \epsilon - (1-c)*\eta$. This is where it gets complicated because now $X^{**}$ is correlated with $\gamma$. OLS will be biased in this case. Please refer to this reference for the details on errors-in-variable analysis.