Statistical matching using fractional imputation 7. Concluding remarksStatistical matching using fractional imputation 7. Concluding remarks

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We approach statistical matching as a missing data problem and propose the PFI method to obtain consistent estimators and corresponding variance estimators. Under the assumption that the specified model is fully identified, the proposed method provides the pseudo maximum likelihood estimators of the parameters in the model.

A sufficient condition for model identifiability is the existence of an instrumental variable in the model. The measurement error framework of Section 5 and Section 6.2, where external calibration provides an independent measurement of the true covariate of interest, is a situation in which the study design may be judged to support the instrumental variable assumption. The proposed methodology is applicable without the instrumental variable assumption, as long as the model is identified. If the model is not identifiable, then the EM algorithm for the proposed PFI method does not necessarily converge. In practice, one can treat the specified model as identified if the EM sequence converges. That is, as long as the EM sequence converges, the resulting analysis is consistent under the specified model. This is one of the main advantages of using the frequentist approach over Bayesian. In the Bayesian approach, it is possible to obtain the posterior values even under non-identified models and the resulting analysis can be misleading.

Testing whether the IV assumption holds in the data at hand is much more difficult, perhaps impossible, under the data structure in Table 1.1. Instead, given the specified model, we can only check whether the model parameters are fully estimable. On the other hand, whether the specified model is a good model for the data at hand is a different story. Model diagnostics and model selection among different identifiable models are certainly important future research topics.

Statistical matching can also be used to evaluate effects of multiple treatments in observational studies. By properly applying statistical matching techniques, we can create an augmented data file of potential outcomes so that causal inference can be investigated with the augmented data file (Morgan and Winship 2007). Such extensions will be presented elsewhere.

Acknowledgements

We thank Professor Yanyuan Ma, an anonymous referee and the Assistant editor (AE) for very constructive comments. The research of the first author was partially supported by Brain Pool program (131S-1-3-0476) from Korean Federation of Science and Technology Society and by a grant from NSF (MMS-121339). The research of the second author was supported by a Cooperative Agreement between the US Department of Agriculture Natural Resources Conservation Service and Iowa State University. The work of the third author was supported by the Bio-Synergy Research Project (2013M3A9C4078158) of the Ministry of Science, ICT and Future Planning through the National Research Foundation in Korea.

Appendix

A. Asymptotic unbiasedness of 2SLS estimator

Assume that we observe $\left(y_1\mathrm{,}x\right)$ in Sample A and observe $\left(y_2\mathrm{,}x\right)$ in Sample B. To be more rigorous, we can write $\left(y_{1a}\mathrm{,}{x}_a\right)$ to denote the observation $\left(y_1\mathrm{,}x\right)$ in Sample A. Also, we can write $\left(y_{2b}\mathrm{,}{x}_b\right)$ to denote the observations in Sample B. In this case, the model can be written as

$$\begin{array}{cc}{y}_{1a}& \mathrm{<mo>=</mo>}{\varphi }_0{1}_a+{\varphi }_1{x}_{1a}+{\varphi }_2{x}_{2a}+e_{1a}\\ y_{2b}& \mathrm{<mo>=</mo>}{\beta }_0{1}_b+{\beta }_1{y}_{1b}+{\beta }_2{x}_{2b}+e_{2b}\end{array}$$

with $E\left(e_{1a}|x_a\right)\mathrm{<mo>=</mo>\; 0}$ and $E\left(e_{2b}|x_b\mathrm{,}{y}_{1b}\right)\mathrm{<mo>=</mo>\; 0.}$ Note that $y_{1b}$ is not observed from the sample. Instead, we use ${\widehat{y}}_{1b}$ using the OLS estimate obtained from Sample A.

Writing $X_a\mathrm{<mo>=</mo>}\left[1_a\mathrm{,}{x}_a\right]$ and $X_b\mathrm{<mo>=</mo>}\left[1_b\mathrm{,}{x}_b\right],$ we have ${\widehat{y}}_{1b}\mathrm{<mo>=</mo>}{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{y}_{1a}\mathrm{<mo>=</mo>}{X}_b{\widehat{\varphi }}_a.$ The 2SLS estimator of $\beta \mathrm{<mo>=</mo>}\left({\beta }_0\mathrm{,}{\beta }_1\mathrm{,}{\beta }_2\right)\prime$ is then

$${\widehat{\beta }}_{2\text{SLS}}\mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}{y}_{2b}$$

where $Z_b\mathrm{<mo>=</mo>}\left[1_b\mathrm{,}{\widehat{y}}_{1b}\mathrm{,}{x}_{2b}\right].$ Thus, we have

$$\begin{array}{ll}{\widehat{\beta }}_{2\text{SLS}}-\beta \hfill & \mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}\left(y_{2b}-Z_b\beta \right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}\left\{{\beta }_1\left(y_{1b}-{\widehat{y}}_{1b}\right)+e_{2b}\right\}\mathrm{.}\hfill \end{array}(\text{A}.1)$$

We may write

$$y_{1b}\mathrm{<mo>=</mo>}{\varphi }_0{1}_b+{\varphi }_1{x}_b+e_{1b}\mathrm{<mo>=</mo>}{X}_b\varphi +e_{1b}$$

where $E\left(e_{1b}|x_b\right)\mathrm{<mo>=</mo>\; 0.}$ Since

$$\begin{array}{ll}{\widehat{y}}_{1b}\hfill & \mathrm{<mo>=</mo>}{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{y}_{1a}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}\left(X_a\varphi +e_{1a}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{X}_b\varphi +X_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{e}_{1a}\mathrm{,}\hfill \end{array}$$

we have

$$y_{1b}-{\widehat{y}}_{1b}\mathrm{<mo>=</mo>}{e}_{1b}-X_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{e}_{1a}$$

and (A.1) becomes

$${\widehat{\beta }}_{2\text{SLS}}-\beta \mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}\left\{{\beta }_1{e}_{1b}-{\beta }_1{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{e}_{1a}+e_{2b}\right\}\mathrm{.}(\text{A}.2)$$

Assume that the two samples are independent. Thus, $E\left(e_{1b}|x_a\mathrm{,}{x}_b\mathrm{,}{y}_{1a}\right)\mathrm{<mo>=</mo>\; 0.}$ Also, $E\left\{{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}{e}_{2b}|x_a\mathrm{,}{x}_b\mathrm{,}{y}_{1a}\mathrm{,}{y}_{1b}\right\}\mathrm{<mo>=</mo>0.}$ Thus,

$$E\left\{{\widehat{\beta }}_{2\text{SLS}}-\beta |x_a\mathrm{,}{x}_b\mathrm{,}{y}_{1a}\right\}\mathrm{<mo>=</mo>}E\left\{-{\beta }_1{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{e}_{1a}|x_a\mathrm{,}{x}_b\mathrm{,}{y}_{1a}\right\}$$

and

$$\begin{array}{ll}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}{X}_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}{e}_{1a}\hfill & \mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}\left\{X_b{\left({X^{ \prime }}_{}{X}_a\right)}^{-1}{X^{ \prime }}_{}\left(y_{1a}-X_a\varphi \right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({Z^{ \prime }}_{}{Z}_b\right)}^{-1}{Z^{ \prime }}_{}{X}_b\left({\widehat{\varphi }}_a-\varphi \right)\mathrm{.}\hfill \end{array}$$

This term has zero expectation asymptotically because $n_b^{-1}{Z^{ \prime }}_{}{Z}_b$ and $n_b^{-1}{Z^{ \prime }}_{}{X}_b$ are bounded in probability and $\left({\widehat{\varphi }}_a-\varphi \right)$ converges to zero.

B. Variance estimation

Let the parameter of interest be defined by the solution to $U_N\left(\eta \right)\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}U\left(\eta \mathrm{<mo>;</mo>}{y}_{1i}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>\; 0.}$ We assume that $\partial U_N\left(\eta \right)/\partial \theta \mathrm{<mo>=</mo>\; 0.}$ Thus, parameter $\eta$ is priori independent of $\theta$ which is the parameter in the data-generating distribution of $\left(x\mathrm{,}{y}_1\mathrm{,}{y}_2\right).$

Under the setup of Section 3, let $\widehat{\theta }\mathrm{<mo>=</mo>}\left({\widehat{\theta }}_1\mathrm{,}{\widehat{\theta }}_2\right)$ be the MLE of $\theta \mathrm{<mo>=</mo>}\left({\theta }_1\mathrm{,}{\theta }_2\right)$ obtained by solving (3.4). Also, let $\widehat{\eta }$ be the solution to $\overline{U}\left(\eta |\widehat{\theta }\right)\mathrm{<mo>=</mo>\; 0}$ where

$$\overline{U}\left(\eta |\theta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ib}{w}_{ij}^{*}U\left(\eta \mathrm{<mo>;</mo>}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{,}$$

and

$$w_{ij}^{*}\propto f\left(y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}|x_i\mathrm{<mo>;</mo>}{\widehat{\theta }}_1\right)f\left(y_{2i}|y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_2\right)/h\left(y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}|x_i\right)$$

with ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij}^{*}\mathrm{<mo>=</mo>\; 1.}$ Here, $h\left(y_1|x\right)$ is the proposal distribution of generating imputed values of $y_1$ in the parametric fractional imputation. By introducing the proposal distribution $h,$ we can safely ignore the dependence of imputed values $y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}$ on the estimated parameter value ${\widehat{\theta }}_1.$

By Taylor linearization,

$$\overline{U}\left(\eta |\widehat{\theta }\right)\cong \overline{U}\left(\eta |\theta \right)+\left(\partial \overline{U}/\partial {{\theta }^{ \prime }}_{}\right)\left({\widehat{\theta }}_1-{\theta }_1\right)+\left(\partial \overline{U}/\partial {{\theta }^{ \prime }}_{}\right)\left({\widehat{\theta }}_2-{\theta }_2\right)$$

Note that

$${\widehat{\theta }}_1-{\theta }_1\cong {\left\{I_1\left({\theta }_1\right)\right\}}^{-1}{S}_1\left({\theta }_1\right)$$

where $I_1\left({\theta }_1\right)\mathrm{<mo>=</mo>}-\partial S_1\left({\theta }_1\right)/\partial {\theta }_{1^{}}.$ Also,

$${\widehat{\theta }}_2-{\theta }_2\cong {\left\{-\frac{\partial }{\partial {{\theta }^{ \prime }}_{}}{\overline{S}}_2\left(\theta \right)\right\}}^{-1}{\overline{S}}_2\left(\theta \right)$$

where

$${\overline{S}}_2\left(\theta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in B}}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_i{w}_{ij}^{*}\left(\theta \right)S_2\left({\theta }_2\mathrm{<mo>;</mo>}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{.}$$

Thus, we can establish

$$\overline{U}\left(\eta |\widehat{\theta }\right)\cong \overline{U}\left(\eta |\theta \right)+K_1{S}_1\left({\theta }_1\right)+K_2{\overline{S}}_2\left(\theta \right)\mathrm{,}$$

where $K_1\mathrm{<mo>=</mo>}{D}_{21}{I}_{11}^{-1}$ and $K_2\mathrm{<mo>=</mo>}{D}_{22}{I}_{22}^{-1}$ with $I_{11}\mathrm{<mo>=</mo>}-E\left(\partial S_1/\partial {{\theta }^{ \prime }}_{}\right),$ $I_{22}\mathrm{<mo>=</mo>}-E\left(\partial {\overline{S}}_2/\partial {{\theta }^{ \prime }}_{}\right),$ $D_{21}\mathrm{<mo>=</mo>}E\left\{U\left(\eta \right)S_1{\left({\theta }_1\right)}^{\prime }\right\}$ and $D_{22}\mathrm{<mo>=</mo>}E\left\{U\left(\eta \right)S_2{\left({\theta }_2\right)}^{\prime }\right\},$ we have

$$V\left\{\overline{U}\left(\eta |\widehat{\theta }\right)\right\}\mathrm{<mo>=</mo>}{\tau }^{-1}\left\{V_1+V_2\right\}{\tau }^{-1^{\prime }}$$

where $\tau \mathrm{<mo>=</mo>}-E\left\{\partial \overline{U}\left(\eta |\theta \right)/\partial {\eta }^{\prime }\right\},$

$$V_1\mathrm{<mo>=</mo>}V\left\{{\displaystyle \sum _{i\in B}}{w}_i\left({\overline{u}}_i^{*}+K_2{S}_{2i}^{*}\right)\right\}\mathrm{,}$$

${\overline{u}}_i^{*}\mathrm{<mo>=</mo>}E\left[U\left(\widehat{\eta }\mathrm{<mo>;</mo>}{y}_{1i}\mathrm{,}{y}_{2i}\right)|y_{2i}\mathrm{<mo>;</mo>}\widehat{\theta }\right],$ and $V_2\mathrm{<mo>=</mo>}V\left\{K_1{\displaystyle {\sum }_{i\in A}}{w}_i{S}_{1i}\right\}\mathrm{.}$ A consistent estimator of each component can be developed similarly to Section 3.

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