Identification of Dynamic Latent Factor Models: The Implications of Re-Normalization in a Model of Child Development

Abstract

A recent and growing area of research applies latent factor models to study the development of children's skills. Some normalization is required in these models because the latent variables have no natural units and no known location or scale. We show that the standard practice of "renormalizing" the latent variables each period is over-identifying and restrictive when used simultaneously with common skill production technologies that already have a known location and scale (KLS). The KLS class of functions include the Constant Elasticity of Substitution (CES) production technologies several papers use in their estimation. We show that these KLS production functions are already restricted in the sense that their location and scale is known (does not need to be identified and estimated) and therefore further restrictions on location and scale by re-normalizing the model each period is unnecessary and over-identifying. The most common type of re-normalization restriction imposes that latent skills are mean log-stationary, which restricts the class of CES technologies to be of the log-linear (Cobb-Douglas) sub-class, and does not allow for more general forms of complementarities. Even when a mean logstationary model is correctly assumed, re-normalization can further bias the estimates of the skill production function. We support our analytic results through a series of Monte Carlo exercises. We show that in typical cases, estimators based on "re-normalizations" are biased, and simple alternative estimators, which do not impose these restrictions, can recover the underlying primitive parameters of the production technology.

Child Care Subsidies with One- and Two-Parent Families

“…This is a higher level of substitutability than what I find, but lower than that in Cunha, Heckman, and Schennach (2010). 7 7 In their 2016 working paper, Agostinelli and Wiswall (2016) the authors show that the assumptions made in Cunha, Heckman, and Schennach (2010) (specifically re-normalization of the latent skill variables) impose over-identifying restrictions which can bias the estimation of the complementarity parameter. The direction of the bias is explored in Monte Carlo simulations which demonstrate that the direction of the bias depends on several attributes of the estimation procedure.…”

Human Capital Development and Parental Investment in India

“…This makes the comparisons over time plausible, as discussed in Agostinelli and Wiswall (2016b). The signal to noise ratios are all above 36%, which shows that most cognitive measures include a substantial amount of information.…”

“…20 The scale of the latent factor is set by the choice of which measurement's factor loading is set to 1, which is salient for the interpretation of the estimates. As pointed out by Agostinelli and Wiswall (2016b), given the longitudinal nature of our model, valid inference across time is only possible if each latent factor is scaled in the same way in every period. One way to meet this condition is to normalize each factor on the same measure every period.…”

Home and School in the Development of Children

“…Note that, at this point, we have not identified the measurement parameters for period 1 (µ j,1,m and λ j,1,m ), only those for period 0. We do not want to impose any restrictions on both periods because these would generally imply restrictions on the skill development process (Agostinelli and Wiswall, 2016b). The equation in (8) can be re-written in "reduced form" as…”

“…Note that we do not impose any restrictions on the latter period (t > 0) latent variables, for example the stock of latent skills in periods t > 0. See(Agostinelli and Wiswall, 2016b) for an analysis the potential biases caused when latent variables are normalized in all periods of dynamic model.…”