Appendix for ”Estimate Non-Finite-Dependent Dynamic Discrete Choice Model with Unobserved Heterogeneity”

Yu (Jasmine) Hao ¹¹1[email protected] HKU Business School, Hong Kong University.
HKU Business School Hiro Kasahara²²2 [email protected] Vancouver School of Economics.
University of British Columbia.

1 Additional Simulation Results

1.1 Two-step Estimator

Table 1: Mean Estimates of Two-step estimator,

X=6250

N=200,T=50

	$\theta^{\mathrm{VP0}}_{m}$	$\theta^{\mathrm{VP1}}_{m}$	$\theta^{\mathrm{VP2}}_{m}$	$\theta^{\mathrm{FC0}}_{m}$	$\theta^{\mathrm{FC1}}_{m}$	$\theta^{\mathrm{EC0}}_{m}$	$\theta^{\mathrm{EC1}}_{m}$	Time	$\rho$
$\gamma_{a}=0$ (finite dependent model)
$\boldsymbol{\theta}^{\mathrm{HM}}$	0.499	1.001	-0.999	0.500	0.998	1.003	0.994	15.740	0.000e+0
	(0.027)	(0.038)	(0.032)	(0.047)	(0.053)	(0.055)	(0.084)	15.350
$\boldsymbol{\theta}^{\mathrm{AFD}}_{1,0}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	1.679	6.317e-6
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	1.240
$\boldsymbol{\theta}^{\mathrm{AFD}}_{1,q^{\mathrm{BE}}}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	32.484	6.317e-6
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	1.240
$\boldsymbol{\theta}^{\mathrm{AFD}}_{2,0}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	3.030	1.037e-11
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	2.470
$\boldsymbol{\theta}^{\mathrm{AFD}}_{3,0}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	4.374	1.720e-17
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	3.679
$\boldsymbol{\theta}^{\mathrm{AFD}}_{4,0}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	5.725	6.938e-19
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	4.905
$\boldsymbol{\theta}^{\mathrm{AFD}}_{5,0}$	0.499	1.001	-0.999	0.501	0.998	1.004	0.991	7.062	6.848e-19
	(0.030)	(0.039)	(0.033)	(0.058)	(0.056)	(0.057)	(0.096)	6.109
$\gamma_{a}=1$ (non-finite dependent model)
$\boldsymbol{\theta}^{\mathrm{HM}}$	0.509	0.998	-0.999	0.523	0.996	0.987	1.008	15.918	0.000
	(0.041)	(0.030)	(0.030)	(0.103)	(0.055)	(0.075)	(0.095)	15.647
$\boldsymbol{\theta}^{\mathrm{AFD}}_{1,0}$	0.416	0.999	-0.999	0.235	0.998	0.988	1.008	1.508	1.024
	(0.094)	(0.031)	(0.032)	(0.286)	(0.059)	(0.077)	(0.106)	1.065
$\boldsymbol{\theta}^{\mathrm{AFD}}_{1,q^{\mathrm{BE}}}$	0.456	0.998	-0.999	0.347	0.997	0.987	1.008	1.633	1.024
	(0.063)	(0.031)	(0.032)	(0.191)	(0.059)	(0.077)	(0.106)	1.065
$\boldsymbol{\theta}^{\mathrm{AFD}}_{2,0}$	0.478	0.997	-0.999	0.412	0.997	0.987	1.008	2.716	0.302
	(0.053)	(0.031)	(0.032)	(0.153)	(0.059)	(0.077)	(0.106)	2.149
$\boldsymbol{\theta}^{\mathrm{AFD}}_{3,0}$	0.491	0.997	-0.999	0.465	0.997	0.987	1.008	3.931	0.075
	(0.050)	(0.031)	(0.032)	(0.135)	(0.059)	(0.077)	(0.106)	3.232
$\boldsymbol{\theta}^{\mathrm{AFD}}_{4,0}$	0.494	0.997	-0.999	0.476	0.997	0.987	1.008	5.149	0.019
	(0.050)	(0.031)	(0.032)	(0.134)	(0.059)	(0.077)	(0.106)	4.313
$\boldsymbol{\theta}^{\mathrm{AFD}}_{5,0}$	0.495	0.997	-0.999	0.478	0.997	0.987	1.008	6.354	0.005
	(0.050)	(0.031)	(0.032)	(0.134)	(0.059)	(0.077)	(0.106)	5.384

1

The data is generated with $\theta=(\theta_{0}^{VP}=0.5,\theta_{1}^{VP}=1,\theta_{2}^{VP}=-1,\theta_{0}^{FC}=0.5,\theta_{1}^{FC}=1.0,\theta_{0}^{EC}=1.0,\theta_{1}^{EC})$ .
2

The first row reports the mean of estimates across 100 Monte Carlo simulations, and the second row reports the standardized mean squared error of the estimates.
3

The second row of the time column with the HM estimator reports the time used for matrix inversion. The second rows of the AFD estimators reports the total time used to solve the optimal weight(weights).

1.2 Sequential Estimator for Two-type Finite Mixture Model

Table 2: Mean Estimator and Squared Bias,

X=15552

, with

N=200,T=50

	Time: 4133.250, Iter: 9.150
		$\theta^{\mathrm{VP0}}_{m}$	$\theta^{\mathrm{VP1}}_{m}$	$\theta^{\mathrm{VP2}}_{m}$	$\theta^{\mathrm{FC0}}_{m}$	$\theta^{\mathrm{FC1}}_{m}$	$\theta^{\mathrm{EC0}}_{m}$	$\theta^{\mathrm{EC1}}_{m}$	$\alpha_{m}$
$\boldsymbol{\theta}^{\mathrm{NPL}}$	Type 1	0.497	1.002	-0.995	0.496	0.992	0.992	0.992	0.403
		(0.061)	(0.045)	(0.046)	(0.144)	(0.126)	(0.081)	(0.149)
	Type 2	1.547	1.035	-1.033	0.588	1.001	0.967	1.057	0.597
		(0.170)	(0.127)	(0.095)	(0.387)	(0.220)	(0.303)	(0.513)
$\boldsymbol{\theta}^{\mathrm{AFD-SEQ}}_{1}$	Type 1	0.454	1.002	-0.994	0.378	0.993	0.992	0.993	0.403
		(0.076)	(0.045)	(0.045)	(0.188)	(0.126)	(0.080)	(0.152)
	Type 2	1.444	1.033	-1.032	0.285	1.002	0.960	1.064	0.597
		(0.171)	(0.126)	(0.093)	(0.419)	(0.220)	(0.307)	(0.518)
	Time: 651.808, Iter: 17.050, $\rho$ : 1.375
$\boldsymbol{\theta}^{\mathrm{AFD-SEQ}}_{2}$	Type 1	0.451	1.002	-0.994	0.370	0.993	0.992	0.993	0.403
		(0.078)	(0.045)	(0.045)	(0.193)	(0.126)	(0.080)	(0.152)
	Type 2	1.442	1.033	-1.032	0.278	1.002	0.960	1.064	0.597
		(0.171)	(0.126)	(0.093)	(0.422)	(0.220)	(0.307)	(0.518)
	Time: 1633.743, Iter: 9.000, $\rho$ : 0.369
$\boldsymbol{\theta}^{\mathrm{AFD-SEQ}}_{3}$	Type 1	0.458	1.002	-0.994	0.400	0.993	0.991	0.993	0.403
	MBE 1	(0.074)	(0.045)	(0.046)	(0.175)	(0.126)	(0.080)	(0.152)
	Type 2	1.459	1.033	-1.032	0.350	1.002	0.961	1.062	0.597
	MBE 2	(0.167)	(0.126)	(0.093)	(0.390)	(0.220)	(0.306)	(0.516)
	Time: 1282.364, Iter: 8.300, $\rho$ : 0.084

1

$nM=200,nT=50$ , based on $20$ Monte Carlo simulations.
2

The true parameters are $\theta_{1}=(\theta_{0}^{VP}=0.5,\theta_{1}^{VP}=1,\theta_{2}^{VP}=-1,\theta_{0}^{FC}=0.5,\theta_{1}^{FC}=1.0,\theta_{0}^{EC}=1.0,\theta_{1}^{EC})=1.0$ , $\theta_{2}=(\theta_{0}^{VP}=1.5,\theta_{1}^{VP}=1,\theta_{2}^{VP}=-1,\theta_{0}^{FC}=0.5,\theta_{1}^{FC}=1.0,\theta_{0}^{EC}=1.0,\theta_{1}^{EC})=1.0$ .

2 Characterization of Weight Solving

2.1 Two periods ”almost finite dependence”

It is possible for us to maximize two-period weight simultaneously so that the objective function becomes

\min_{\mathbf{w}_{1},\mathbf{w}_{2}}\lVert\mathbf{\tilde{F}}\mathbf{F}(\mathbf{w}_{1})\mathbf{F}(\mathbf{w}_{2})\rVert

Note that $\sum_{d^{\dagger}\in\mathcal{D}}\mathbf{w}_{1}(d^{\dagger},i^{\dagger})f(i^{\dagger\dagger}|d^{\dagger},i^{\dagger})=\sum_{d^{\dagger}\in\mathcal{D}/\{0\}}\mathbf{w}_{1}(d^{\dagger},i^{\dagger})\tilde{f}(i^{\dagger\dagger}|d^{\dagger},i^{\dagger})+f(i^{\dagger\dagger}|0,i^{\dagger})$ . The $(d,i,j)$ -th element of the objective function, which is the $(ij)$ -th element of $\mathbf{\tilde{F}}(d)\mathbf{F}(\mathbf{w}_{1})\mathbf{F}(\mathbf{w}_{2})$ is

\begin{split}&[\mathbf{\tilde{F}}(d)\mathbf{F}(\mathbf{w}_{1})\mathbf{F}(\mathbf{w}_{2})]_{ij}\\ &=\sum_{\begin{subarray}{c}i^{\dagger}\in\mathcal{X},\\ i^{\dagger\dagger}\in\mathcal{X}\end{subarray}}\tilde{f}(i^{\dagger}|d,i)\underbrace{\left(\sum_{d^{\dagger}\in\mathcal{D}/\{0\}}\mathrm{w}^{1}(d^{\dagger},i^{\dagger})\tilde{f}(i^{\dagger\dagger}|d^{\dagger},i^{\dagger})+f(i^{\dagger\dagger}|0,i^{\dagger})\right)}_{f^{\mathbf{w}_{1}}(i^{\dagger\dagger}|i^{\dagger})}\\ &\times\underbrace{\left(\sum_{d^{\dagger\dagger}\in\mathcal{D}/\{0\}}\mathrm{w}^{2}(d^{\dagger\dagger},i^{\dagger\dagger})\tilde{f}(j|d^{\dagger\dagger},i^{\dagger\dagger})+f(j|0,i^{\dagger\dagger})\right)}_{f^{\mathbf{w}_{2}}(j|i^{\dagger\dagger})}\\ &=(\mathbf{w}_{1}^{+})^{\top}\mathbf{H}(d,i,j)\mathbf{w}_{2}^{+}+(\mathbf{w}_{1}^{+})^{\top}\mathbf{h^{1}}(d,i,j)+(\mathbf{w}_{2}^{+})^{\top}\mathbf{h^{2}}(d,i,j)+\mathbf{h^{0}}(d,i,j).\end{split}

(1)

where $\mathbf{w}_{1}^{+}=[\mathrm{w}^{1}(1,1)\ldots,\mathrm{w}^{1}(1,X),\ldots,\mathrm{w}^{1}(D,1)\ldots,\mathrm{w}^{1}(D,X)]^{\top}$ and
$\mathbf{w}_{2}^{+}=[\mathrm{w}^{2}(1,1)\ldots,\mathrm{w}^{2}(1,X),\ldots,\mathrm{w}^{2}(D,1)\ldots,\mathrm{w}^{2}(D,X)]^{\top}$ are $\big{(}DX\big{)}\times 1$ vectors such that for each state $x$ , we leave $w(0,x)$ out.

\begin{split}\underbrace{\mathbf{H}(d,i,j)}_{(XD)\times(XD)\text{ matrix }}&=\Big{[}\tilde{f}(i^{\dagger}|d,i)\tilde{f}(i^{\dagger\dagger}|d^{\dagger},i^{\dagger})\tilde{f}(j|d^{\dagger\dagger},i^{\dagger\dagger})\Big{]}_{\big{(}d^{\dagger},i^{\dagger}\big{)},\big{(}d^{\dagger\dagger},i^{\dagger\dagger}\big{)}},\\ \underbrace{\mathbf{h^{1}}}_{(XD)\times 1\text{ row vector }}&=\Big{[}f(i^{\dagger}|d,i)\sum_{i^{\dagger\dagger}\in\mathcal{X}}\tilde{f}(i^{\dagger\dagger}|d^{\dagger},i^{\dagger})f(j|0,i^{\dagger\dagger})\Big{]}_{\big{(}d^{\dagger},i^{\dagger})\big{)}},\\ \underbrace{\mathbf{h^{2}}}_{(XD)\times 1\text{ row vector }}&=\Big{[}\sum_{i^{\dagger}\in\mathcal{X}}f(i^{\dagger}|d,i)f(i^{\dagger\dagger}|0,i^{\dagger})\tilde{f}(j|d^{\dagger\dagger},i^{\dagger\dagger})\Big{]}_{\big{(}d^{\dagger\dagger},i^{\dagger\dagger})\big{)}},\\ \mathrm{h}^{0}&=\sum_{\begin{subarray}{c}i^{\dagger}\in\mathcal{X},\\ i^{\dagger\dagger}\in\mathcal{X}\end{subarray}}f(i^{\dagger}|d,i)f(i^{\dagger\dagger}|0,i^{\dagger})\tilde{f}(j|0,i^{\dagger\dagger}).\end{split}

(2)

Note that a 1-period finite dependence model is equivalent to the condition that that there exists $\mathbf{w}_{1}^{+},\mathbf{w}_{2}^{+}$ such that:

\sum_{\begin{subarray}{c}d\in\mathcal{D}/\{0\},\\ i\in\mathcal{X},\\ j\in\mathcal{X}\end{subarray}}\left((\mathbf{w}_{1}^{+})^{\top}\mathbf{H}(d,i,j)\mathbf{w}_{2}^{+}+(\mathbf{w}_{1}^{+})^{\top}\mathbf{h^{1}}(d,i,j)+(\mathbf{w}_{2}^{+})^{\top}\mathbf{h^{2}}(d,i,j)+\mathbf{h^{0}}(d,i,j)\right)^{2}=0.

With the above expression, the objective of the minimization is

\begin{split}\min_{\mathbf{w}_{1}^{+},\mathbf{w}_{2}^{+}}\lVert\tilde{\mathbf{F}}\mathbf{F}(\mathbf{w}^{1})\mathbf{F}(\mathbf{w}^{2})\rVert&=\min_{\mathbf{w}_{1}^{+},\mathbf{w}_{2}^{+}}\sum_{\begin{subarray}{c}d\in\mathcal{D}/\{0\},\\ i\in\mathcal{X},\\ j\in\mathcal{X}\end{subarray}}\Big{(}(\mathbf{w}_{1}^{+})^{\top}\mathbf{H}(d,i,j)\mathbf{w}_{2}^{+}\\ &+(\mathbf{w}_{1}^{+})^{\top}\mathbf{h^{1}}(d,i,j)+(\mathbf{w}_{2}^{+})^{\top}\mathbf{h^{2}}(d,i,j)+\mathbf{h^{0}}(d,i,j)\Big{)}^{2}.\end{split}