Possible to compare significance levels of coefficient estimates using some test?

[Tomassi]

Hi guys! I have got a question and was hoping some bright guy here could help

me out;

I ran two regressions:

Yt = c + [beta1Y]Xt + [beta2Y]Xt-1 + [beta3Y]Ct + [beta4Y]Yt-1

Zt = c + [beta1Z]Xt + [beta2Z]Xt-1 + [beta3Z]Ct + [beta4Z]Zt-1

Yt is the trading volume generated by 'investor group 1' and Zt is the

trading volume generated by 'investor group 2'. Xt is the market return. Xt-1,

Ct and both Yt-1 and Zt-1 serve as control variables. Both beta1Y and beta1Z

are significant.

Now I would like to test whether beta1Y is -so to speak- 'more significant'

than beta1Z. (Essentially I want to test whether investor group 1 is more

'sensitive' to the contemporaneous market return when trading, than investor

group 2)

For each of the above beta's I have obtained coefficient estimates, standard

errors and (as a result also) t-statistic values. I was thinking to use a t-

test and do the following (taking into account that the standard error of a t-

statistic value is 1,645):

t = (t.statistic.value[beta1Y] - t.statistic.value[beta1Z])/SQRT([1,645^2]/n+

[1,645^2]/n)

If t > 1,645 I would conclude beta1Y is indeed more significant than beta1Z.

But I am not sure if this is statistically speaking an acceptable test? Is

what I aiming to do 'OK'? Am I using the correct formula? Or should I use

another test? (if so, which one.. ?)

Thanks a lot for your insights here, any help is truly appreciated!

[sisyphus1]

you would need to get var(betaY - betaZ), for which youl would need their covariance. covariance won't be 0 because they both have X's in them, but you should be able to figure out the covariance from the formulas of betaY-betaZ

[Tomassi]

Hi sisyphus1, thanks a lot for your reply.

I am not sure though I understand what you believe is the proper approach for testing whether "investor group 1 is more 'sensitive' to the contemporaneous market return when trading, than investor group 2"

beta1Y and beta1Z are regression result coefficient estimates, not time-series variables. I only got one value for each of them, and as far as I can see I cannot determine a variance for the variable 'beta1Y-beta1Z'. (?)

ps. A final possibly relevant note; if this was not already clear from the OP; these are time series regressions. Both Yt and Zt are regressed on exactly the same set of Xt (and control variable) values.

Edited November 2, 2011 by Tomassi

[sisyphus1]

beta1y and beta1z are have formulas (as a function of X's and Y's for beta1y and of X's and Z's). since they share the X's, we know for sure that their cov is not 0. (not sure if you will need cov(Y,Z) for this as well, but you should be able to get this from your data).

[Tomassi]

Ok, not sure if I correctly followed up on your suggestions, but this is what I have done:

Using the regression coefficient estimates that I obtained for c, beta1Y, beta2Y, beta3Y and beta4Y (obtained by running the regression "Yt = c + [beta1Y]Xt + [beta2Y]Xt-1 + [beta3Y]Ct + [beta4Y]Yt-1 + errorterm"), I 'reverse engineered' a beta1Y-value for each time t. Thus, for each t I determined:

[beta1Y]_t = (Y_t - (c + [beta2Y.regressionestimate]*X_t-1 + [beta3Y.regressionestimate]*C_t + [beta4Y.regressionestimate]*Y_t-1)) / X_t

Similarly, I determined [beta1Z]_t-values for each t, resulting in having two time series; of beta1Y and beta1Z. Now the variance of (beta1Y - beta1Z) can straightforwardly be computed in Excel by first creating a new time series: "beta1Y - beta1Z", but alternatively I can indeed also do what -I suppose- is what you have been suggesting: seperately determine the variance for the beta1Y and the beta1Z time series, as well as the covariance for the beta1Y and beta1Z time series, and subsequenlty compute the variance for (beta1Y - beta1Z) by using the formula: VAR(beta1Y)+VAR(beta1Z)-2*COVAR(beta1Y,beta1Z). These two approaches yield almost identical variance.of[beta1Y-beta1Z] values.

Should I now execute the t-test as:

t= (mean[beta1Y.timeseries] - mean[beta1Z.timeseries]) / SQRT(variance.of[beta1Y - beta1Z])

Or should I maybe compute:

t = (beta1Y.regressionestimate - mean[beta1Z.regressionestimate]) / SQRT(variance.of[beta1Y - beta1Z])

Or something else? (not sure exactly what t-test formula to use now..)

Neither of the above test formulae yield a t-test value that is anything close to significant. Intuitively this seems very unlikely for my sample as the t-statistic values reported for the beta1Y.regressionestimate equals 3,160 and for the beta1Z.regressionestimate 1,720. Intuitively I would say beta1Y must thus be (at least close to being) 'significantly more significant' than beta1Z.

(Consequently I would suppose I am ought to use another t-test formula, or that I am simply making some mistake; possibly I am forgetting to divide/multiply by n somewhere in the above formulae?)

Hopefully you're still following me? Please tell me if I need to clarify anything.

Again, your insights and help are really appreciated.. Many, many thanks

Edited November 2, 2011 by Tomassi