2SLS tobit and xtpoisson

[Francesco DAlessandro]

Dear Statalisters, I am struggling to build a 2 step estimation procedure.

In particular, as a dependent variable, I have the count of new entrants at the regional level (nuts-2) from 1990 to 2020. I am interested in the effect of two regressors on my dependent variable: on the one hand, the knowledge stock at the nuts-2 level (proxied by a fractional counting of inventors for each patent; therefore it is a continuous variable and not a proper count variable) and, on the other hand, R&D expenditures aggregated at the nuts-2 level. I have estimated a panel poisson model with fixed effects "xtpoisson new_entrants L1.k_stock L1.R&D L1.controls, fe", following Wooldridge (1999), hence avoiding the panel specification of the negative binomial with fe.

However, as you may imagine, R&D expenditures and knowledge stock are highly correlated, therefore including both regressors in the same model would produce multicollinearity issues. Therefore, I would like to implement a first step where: the knowledge stock is regressed on the lagged R&D expenditures (y=knowledge stock at time t, x=R&D expenditures at time t-1) and then take the residuals. Then, I would estimate my main model again by substituting the knowledge stock with the residuals of the first step (which can be interpreted as the amount of knowledge stock NOT explained by R&D expenditures undertaken by large corporations, but due to serendipitous inventions of individuals), as: "xtpoisson new_entrants L1.res L1.controls, fe".

In this respect, as a FIRST STEP, I considered estimating a tobit model since the knowledge stock varies from 0 to 860.35 (many corner solutions occur). Therefore, "tobit k_stock L1.R&D, nocons ll(0)" or even "xttobit k_stock L1.R&D, nocons ll(0)".

Do you know how I can proceed from here? Do you have any suggestions?

[Francesco DAlessandro]

What do you think about using a normal OLS as a first step and then getting the residuals? When I estimate the tobit model as tobit k_stock L1.R&D, nocons ll(0) and predict the fitted values for my dependent variable as predict k_stock_h, ystar(0,.), i get very strange results that are very high compared to those I get from OLS implementing these two lines of commands: reg k_stock L1.R&D, nocons and predict k_stock_h, xb. Below a table of the results; k_ai_h_reg are the fitted values from OLS, while k_stock_h_tob are the fitted values from Tobit:

nuts_2	year	R&D	k_stock	k_stoch_h_reg	k_stock_h_tob
AT12	1990	12.75	0	.	.
AT12	1991	21.79	0	0.44	9.71
AT12	1992	24.54	0	0.76	9.88
AT12	1993	29.24	0	0.86	9.93
AT12	1994	42.50	0	1.02	10.01
AT12	1995	60.49	0	1.49	10.25
AT12	1996	63.55	0.91	2.12	10.59
AT12	1997	82.98	1.29	2.23	10.65
AT12	1998	101.92	1.12	2.91	11.02

I don't understand the results i get from Tobit since, when estimating the marginal effect as margins, dydx(L1.k_ict) predict(ystar(0,.)) at(L1.k_ict=(0(1)10)) i get a marginal effect at L1.k_ict=0 of 0.01756 and at L1.k_ict=1 of 0.01758. Is this difference due to non-linearity?

To workaround, I have tried to estimate one partial effect for each level of the variable R&D in the sample following these lines of code:

tobit k_stock L1.R&D, nocons ll(0)
predict xbh, xb
gen sigma=sqrt(_b[/var(e.k_stock)])
gen scale=normal(xbh/sigma)
gen pe_R&D=scale*_b[L1.R&D]

Now, for each observation, I have a partial effect:

nuts_2	year	R&D	pe_R&D	k_stock
AT12	1990	12.75	.	0
AT12	1991	21.79	0.017831	0
AT12	1992	24.54	0.0180179	0
AT12	1993	29.24	0.0180746	0
AT12	1994	42.50	0.0181721	0
AT12	1995	60.49	0.0184462	0
AT12	1996	63.55	0.0188174	0.91
AT12	1997	82.98	0.0188804	1.29
AT12	1998	101.92	0.01928	1.12

Is it wrong to compute the fitted values as gen k_stock_h=L1.R&D*pe_R&D? I suppose so due to non-linearity...