Inconsistencies in estimating parameters in a simple Dynamic Panel Data problem using ivregress, xtabond and xtdpdml

[Adriaan Hoogendoorn]

While having only two variables of patients across time, I got lost into dynamic panel data modelling. ‘How hard can it be?’, I thought originally.

I use data on scores of both physical health and mental health of patients at four different times and intend to do cross-lagged regression analysis. Being a newbie in this field I used Rabe-Hesketh & Skrondal’s (RH&S’s) book on Multilevel and Longitudinal Modeling Using Stata (2012) as a guideline.

Having declared the panel data and using Stata’s easy way of referring to lagged variables, the naïve way would be

xtset id t
regress mental L.mental L.physical

Since (repeated) observations are nested within patients, residuals are expected to be correlated, and I became tempted to use

mixed mental L.mental L.physical || id: ,  res(uns, t(t)) reml

RH&S state that this is be problematic, since “it would produce inconsistent estimates of the regression coefficients because lagged responses included as covariates, are correlated with the random intercept” (p273).

The authors suggest two ways out: 1. the Anderson-Hsiao approach and 2. the Arrelano-Bond approach.

In the Anderson-Hsiao approach, the second lag of the responses is used as instrumental variable for the lagged difference. Translating their example (p275) to our situation I used:

ivregress 2sls D.mental LD.(physical) (LD.mental=L2.mental)

For the Arrelano-Bond approach (p277) I used:

xtabond mental L.physical , twostep noconstant vce(robust)

Additionally, as a third method, I used the xtdpdml command by Allison, Moral-Benito and Williams (2015)

xtdpdml mental L.physical

The regression parameters that I found for L.mental (with SE) and L.physical (with SE) were respectively:

regress ______ 0.74 (0.06) 0.01 (0.01)
mixed ________ 0.94 (0.04) 0.00 (0.01)
ivregress ___ -0.58 (0.34) 0.01 (0.04)
xtabond ______ 0.40 (0.50) 0.09 (0.05)
xtdpdml ______ 0.30 (0.24) 0.06 (0.03)

The results clearly illustrate how the two naïve methods differ from the latter three methods. Somehow I hoped that the results of the latter three methods would coincide. Unfortunately, they do not. The result that puzzles me most is the deviant estimate for L.mental coming from the ivregress method (notice the minus sign). In fact, in the Stata output of ivregress the parameter is named LD.mental - not L.mental, but RH&S state it estimates the same parameter “gamma” (Table 5.2, p272). My question is: Are there any Stata users that experienced similar differences in estimates from ivregress and xtabond? Am I missing something?

Kind regards,
Adriaan Hoogendoorn


References
Rabe-Hesketh, Sophia & Anders Skrondal (2012) Multilevel and Longitudinal Modeling Using Stata (Third edition), Stata Press.
Allison, Moral-Benito and Williams (2015) https://ideas.repec.org/p/boc/scon15/11.html

[Herman vandenBerg]

Adriaan,

I'm also a newbie, but I understand that the differences in the results are a consequence of the different assumptions inherent in each estimator.

In my latest research, I used "xtdpdml" instead of "xtabond" because the "xtdpdml" uses a maximum likelihood estimator implemented with structural equation modeling, and this is substantially more efficient in handling dynamic panel data than the generalized method of moments used with "xtabond" when the normality assumption is met and "xtdpdml" suffers less from finite sample biases (Hsiao, Pesaran, and Tahmiscioglu, 2002; Williams et al., 2015).

The GMM method can also perform poorly when the autoregressive parameter is close to 1 (Blundell, Bond, and Windmeijer, 2001; Williams et al., 2015), which was the case with my measures of the dependent variable.

Finally, my model had all four types of independent variables, which the “xtdpdml” estimator could accommodate: lags of the dependent variable; strictly exogenous variables that (by assumption) are uncorrelated with the error terms at all points in time; predetermined variables, also known as sequentially exogenous variables, that can be affected by prior values of the dependent variable; and time-invariant exogenous variables whose values are constant across time (Williams et al., 2015).

I hope that helps a bit.

Best,
Herman van den Berg

[Enrique Pinzon (StataCorp)]

Hello Adriaan,

I would like to comment on your question about ivregress and xtabond . The biggest difference between the Anderson-Hsiao estimator, which you computed using ivregress, and the Arellano-Bond estimator are the set of instruments used. For your case, you are using the second lag of the dependent variable, mental, in the Anderson-Hsiao case. Arellano-Bond uses all available lags starting with the second lag. Therefore, the instrument set is much larger and this is bound to generate different point estimates. A while back I wrote a post about this that might prove to be useful to understand where the Arellano-Bond instruments come from.

http://blog.stata.com/tag/arellano-bond/

[Richard Williams]

Let me just say I am pleased that Herman and Adriaan are using xtdpdml even though we haven't officially released it yet! And I am also pleased that xtdpdml and xtabond are giving similar results. Hopefully xtdpdml's are better.

We plan on sending the program to SSC soon. In the meantime, the program's support page (which includes information about the method and how to download the beta version) is at

http://www3.nd.edu/~rwilliam/dynamic/index.html

We have a couple of papers in draft form and we will be adding them to the support page soon.