Identification and semiparametric estimation of a finite horizon dynamic discrete choice model with a terminating action

Abstract

We study identification and estimation of finite-horizon dynamic discrete choice models with a terminal action. We first demonstrate a new set of conditions for the identification of agents’ time preferences. Then we prove conditions under which the per-period utilities are identified for all actions in the agent’s choice-set, without having to normalize the utility for one of the actions. Finally, we develop a computationally tractable semiparametric estimator. The estimator uses a two-step approach that does not use either backward induction or forward simulation. Our methodology can be implemented using standard statistical packages without the need to write specialized computational routines, as it involves linear (or nonlinear) projections only. Monte Carlo studies demonstrate the superior performance of our estimator compared with existing two-step estimation methods. Monte Carlo studies further demonstrate that the ability to identify the per-period utilities for all actions is crucial for counterfactual predictions. As an empirical illustration, we apply the estimator to the optimal default behavior of subprime mortgage borrowers, and the results show that the ability to identify the discount factor, rather than assuming an arbitrary number as typically done in the literature, is also crucial for obtaining correct counterfactual predictions. These findings highlight the empirical relevance of key identification results of the paper.

Appendices

Appendix A: Optimal policy functions

Proposition 1

Under Assumption 1 there exists a unique decision rule $D_{t}^{\ast }(s_{t},\varepsilon _{t})$ supported on A for t=1,2,…,T that solves the maximization problem

$$\sup\limits_{(D_{1},D_{2},\ldots,D_{T})\in A^{T}}V_{1,\sigma}(s_{1}). $$

Proof 3

Our argument uses backward induction. In the final period (at mortgage maturity) the borrower faces a static optimization problem of choosing among V _T (0,s _T ) + ε _0,T , V _T (1,s _T ) + ε _1,T , and V _T (2,s _T ) + ε _2,T . The optimal decision delivers the highest payoff, yielding the decision rule $D_{T}^{\ast }(s_{T},\varepsilon _{T})=\text {arg\,max}_{k\in A}\{V_{T}(k,s_{T}) + \varepsilon _{k,T}\}$. Provided that the payoff shocks are idiosyncratic and have a continuous distribution, the optimal choice probabilities are characterized by continuous functions of (V _T (k,s _T ), k∈A). Knowing the optimal decision rule in period T, we can obtain the choice-specific value function in period T−1 as

$$\begin{array}{@{}rcl@{}} \textstyle V_{T-1}(k,s_{T-1})&=&u(k,s_{T-1})+\beta E\left[ \sum\limits_{k^{\prime}\in A}\mathbf{1}\{D_{T}^{\ast}=k^{\prime}\}\left( V_{T}(k^{\prime},s_{T})+\varepsilon_{k^{\prime},T}\right) \right.\\ &&\quad\quad\quad\quad\quad\quad\quad\quad\left.\big\vert\,s_{T-1},a_{T-1}=k\vphantom{\sum\limits_{k^{\prime}\in A}}\right] . \end{array} $$

Provided that the T ^th period optimal decision has already been derived, the optimal decision problem in T−1 becomes a static choice among three alternatives. Its solution, again, trivially exists and is (almost surely) unique because the distribution of ε _T−1 is continuous. We iterate this procedure back to t = 1. □

Appendix B: Lemma 1

Under our assumptions, the system of equations

$$\begin{array}{@{}rcl@{}} && \sigma_{0}(z_{1},z_{2})=\bar{\sigma}_{0},\\ && \sigma_{1}(z_{1},z_{2})=\bar{\sigma}_{1} \end{array} $$

has a unique solution if and only if $\bar {\sigma }_{0}+\bar {\sigma }_{1}<1$.

This result generalizes that in Hotz and Miller (1993) to general full support distributions, and is also proved in Norets and Takahashi (2013). For completeness of exposition, we provide the proof.

Proof 4

Consider partial derivatives

$$\begin{array}{@{}rcl@{}} \frac{\partial\sigma_{0}(z_{1},z_{2})}{\partial z_{1}} & =&-{\int}_{-\infty }^{+\infty}\frac{\partial^{2}F_{\varepsilon}}{\partial\varepsilon_{0} \partial\varepsilon_{1}}(\varepsilon_{0},\varepsilon_{0}-z_{1},\varepsilon_{0}-z_{2})\,d\varepsilon_{0},\;\;\\ \frac{\partial\sigma_{0}(z_{1},z_{2})}{\partial z_{2}} & =&-{\int}_{-\infty }^{+\infty}\frac{\partial^{2}F_{\varepsilon}}{\partial\varepsilon_{0} \partial\varepsilon_{2}}(\varepsilon_{0},\varepsilon_{0}-z_{1},\varepsilon_{0}-z_{2})\,d\varepsilon_{0}. \end{array} $$

Similarly, we can find that

$$\begin{array}{@{}rcl@{}} \frac{\partial\sigma_{1}(z_{1},z_{2})}{\partial z_{1}}&= & {\int}_{-\infty }^{+\infty}\frac{\partial^{2}F_{\varepsilon}}{\partial\varepsilon_{0} \partial\varepsilon_{1}}(z_{1}+\varepsilon_{1},\varepsilon_{1},z_{1} -z_{2}+\varepsilon_{1})\,d\varepsilon_{1}\\ && +{\int}_{-\infty}^{+\infty}\frac{\partial^{2}F_{\varepsilon}}{\partial \varepsilon_{1}\partial\varepsilon_{2}}(z_{1}+\varepsilon_{1},\varepsilon_{1},z_{1}-z_{2}+\varepsilon_{1})\,d\varepsilon_{1}\\ & =&{\int}_{-\infty}^{+\infty}\left( \frac{\partial^{2}F_{\varepsilon} }{\partial\varepsilon_{0}\partial\varepsilon_{1}}+\frac{\partial ^{2}F_{\varepsilon}}{\partial\varepsilon_{1}\partial\varepsilon_{2}}\right) (\varepsilon_{0},\varepsilon_{0}-z_{1},\varepsilon_{0}-z_{2})\,d\varepsilon_{0}, \end{array} $$

and

$$\frac{\partial\sigma_{1}(z_{1},z_{2})}{\partial z_{2}}=-{\int}_{-\infty }^{+\infty}\frac{\partial^{2}F_{\varepsilon}}{\partial\varepsilon_{1} \partial\varepsilon_{2}}(\varepsilon_{0},\varepsilon_{0}-z_{1},\varepsilon_{0}-z_{2})\,d\varepsilon_{0}. $$

We assumed that the joint distribution of errors has a continuous density with a full support on ${\mathbb {R}}^{3}$. Provided that $\frac {\partial \sigma _{0}(z_{1},z_{2})}{\partial z_{1}}\,\frac {\partial \sigma _{0}(z_{1},z_{2} )}{\partial z_{2}}>0$ the mapping z ₁↦z ₂ implicitly defined by equation $\sigma _{0}(z_{1},z_{2})=\bar {\sigma }_{0}$ is invertible. Moreover, if we denote this mapping $z_{2}=m_{0}(z_{1},\bar {\sigma }_{0})$, then using the result regarding the derivative of the inverse function, we can conclude that

$$\frac{\partial m_{0}(z_{1},\bar{\sigma}_{0})}{\partial z_{1}}\leq0. $$

Similarly, we can define a map $z_{2}=m_{1}(z_{1},\bar {\sigma }_{1})$, then using the result regarding the derivative of the inverse function, we can conclude that

$$\frac{\partial m_{1}(z_{1},\bar{\sigma}_{1})}{\partial z_{1}}\geq0. $$

We can explore the asymptotic behavior of both maps. Consider m ₀ first. Suppose that z ₁→−∞. Then $\lim \limits _{z_{1} \rightarrow -\infty }m_{0}(z_{1},\bar {\sigma }_{0})=z_{2}^{\ast }$, where $z_{2}^{\ast }$ solves $\int \mathbf {1}\{\varepsilon _{0}\geq z_{2}^{\ast }+\varepsilon _{2}\}F_{\varepsilon }(d\varepsilon \,)=\bar {\sigma }_{0}$. Also let $z_{1}^{\ast }$ solve $\int \mathbf {1}\{\varepsilon _{0}\geq z_{1}^{\ast }+\varepsilon _{1}\}F_{\varepsilon }(d\varepsilon \,)=\bar {\sigma }_{0}$. Then $\lim \limits _{z_{1}\rightarrow z_{1}^{\ast }}m_{0}(z_{1},\bar {\sigma }_{0})=-\infty$.

Next consider m ₁. Suppose that $z_{2}^{\ast \ast }$ is the solution of $\int \mathbf {1}\{\varepsilon _{1}\geq z_{2}^{\ast \ast }+\varepsilon _{2}\}F_{\varepsilon }(d\varepsilon )=\bar {\sigma }_{1}$. Then as z ₁→ + ∞, the map approaches asymptotically to the line: $m_{1}(z_{1},\bar {\sigma }_{1})\rightarrow z_{1}+z_{2}^{\ast \ast }$. Suppose that $z_{1}^{\ast \ast }$ is the solution of $\int \mathbf {1}\{z_{1}^{\ast \ast }+\varepsilon _{1}\geq \varepsilon _{0}\}F_{\varepsilon }(d\varepsilon \,)=\bar {\sigma }_{1}$. Then $\lim \limits _{z_{1}\rightarrow z_{1}^{\ast \ast } }m_{1}(z_{1},\bar {\sigma }_{1})=-\infty$. Thus m ₀ is a continuous strictly decreasing mapping from $(-\infty ,\,z_{1}^{\ast }]$ into $(-\infty ,z_{2}^{\ast }]$ and m ₁ is a continuous strictly increasing mapping from $[z_{1}^{\ast \ast },+\infty )$ into the real line.

Provided that both curves are continuous and monotone, they intersect if and only if their projections on z ₁ and z ₂ axes overlap. The projections on the z ₂ axis are guaranteed to overlap ($(-\infty ,\,z_{2}^{\ast }]\subset {\mathbb {R}}$). The projections on the z ₁ axis will overlap if and only if $z_{1}^{\ast \ast }<z_{1}^{\ast }$. Given that function $\sigma (z)=\int \mathbf {1}\{\varepsilon _{0}-\varepsilon _{1}\leq z\}F_{\varepsilon }(d\varepsilon \,)$ is strictly monotone in z, then $z_{1}^{\ast \ast }<z_{1}^{\ast }$ if and only if $\bar {\sigma }_{0}+\bar {\sigma }_{1}<1$. This proves the statement of Lemma 1. □

Appendix C: Asymptotic theory for the plug-in estimator

Section 3 outlined the structure of the two-step plug-in estimator for the structural parameters, which include the per-period payoffs and the discount factor. This Appendix provides the asymptotic theory for the constructed estimator. We assume a parametric specification for the per-period utility, although our theory allows for an immediate extension to a nonparametric specification of the per-period utility. A key requirement of the plug-in semiparametric procedure is that the first-stage nonparametric estimator of the policy functions converge at a sufficiently fast rate. Our results for the consistency and the convergence rate of the first-stage estimator rely on the results in Wong and Shen (1995), Andrews (1991), and Newey (1997).

To assure consistency and a fast convergence rate for the first-stage estimator, we need the following assumption.

Assumption 2

• (i) In addition to the Markov assumption (Assumption 1.iii), for each period t the distribution of states s _t |s _t−1 is identical across borrowers and over time, and the choice probabilities σ _k,t (⋅) are uniformly bounded from 0 and 1 for each k=0,1,2. The state space ${\mathcal {S}}$ is compact.

• (ii) The eigenvalues of E[q ^L (s _t )q ^L′ (s _t ) | a _t ] are bounded away from zero uniformly over L, and |q _l (⋅)|≤C for all l.

• (iii) $\frac {\sigma _{k,t}(s)}{\sigma _{0,t}(s)}$ belongs to a separable functional space with basis $\{q_{l}(\cdot )\}_{l=1}^{\infty }$ . For each t≤T and k∈{1,2} the selected series terms provide a uniformly good approximation for the probability ratio

  $$\sup\limits_{s\in{\mathcal{S}}}\left\Vert \log\,\frac{\sigma_{k,t}(s)} {\sigma_{0,t}(s)}-\text{proj}\left( \log\,\frac{\sigma_{k,t}(s)}{\sigma_{0,t}(s)}\,\big\vert\,q^{L}(\cdot)\right) \right\Vert =O(L^{-\alpha}) $$

  for some $\alpha \geq \frac {1}{2}$.

Assumption 2 can be verified for particular classes of polynomials and sieves (see Chen 2007). Assumption 2 implies the following result establishing the consistency and convergence rate of the first-stage estimator for the policy functions.³²

Theorem 3

Under Assumptions 1 and 2, the estimator (4) is consistent uniformly over s:

$$\sup\limits_{s\in{\mathcal{S}}}\left\Vert \widehat{\sigma}_{k,t}(s)-{\sigma }_{k,t}(s)\right\Vert =o_{P}\left( J^{-1/4}\right) $$

provided that L→∞ with $\frac {J}{L\log (J)}\rightarrow \infty$ as J→∞.

The asymptotics in this theorem is in terms of the number of loans J, reflecting the fact that each loan is observed only once for a given t. We use the estimated first-stage policy functions as inputs for the estimation of the second-stage structural parameters. Our approach is based on applying existing plug-in implementations for estimating the system of Eq. 5. These techniques involve constructing nonparametric elements based on a statistical model (in our case, the policy functions) that are then plugged into a fully parametric second step. Estimation in the second step is commonly performed by means of a weighted minimum distance procedure, with weights that are chosen optimally to maximize the efficiency of the resulting estimator.

To establish the asymptotic properties of the designed procedure we impose the following assumptions.

Assumption 3

• (i) Parameter space Θ is a compact subset of ${\mathbb {R}}^{p}$.

• (ii) The per-period payoff is Lipschitz-continuous in parameters.

• (iii) The variance of the one-period-ahead policy function is bounded ($\sup \limits _{s\in {\mathcal {S}}}E\left [ \sigma _{k,t+1}(s_{t+1} )^{2}\,|\,s_{t}=s\right ] <1$ ) and strictly positive ( $\inf \limits _{s\in {\mathcal {S}}}E\left [ \sigma _{k,t+1}(s_{t+1})^{2}\,|\,s_{t}=s\right ] >0$ ) for any t<T.

Under this assumption and the technical assumption described in Appendix Appendix, which restricts the complexity of the class of functions that is associated with our “nonparametric multinomial logit” estimator, we can use the results regarding semiparametric plug-in estimators in Ai and Chen (2003) and Chen et al. (2003), and establish the following result for the estimator for the second-stage structural parameters.

Theorem 4

Under Assumptions 1, 2 and 3, the estimator (5) is consistent and has asymptotic normal distribution:

$$\sqrt{JT^{\ast}}\left( \left( \hat{\theta}(a),\hat{\beta}\right) -\left( \theta_{0}(a),\beta\right) \right) \overset{d}{\longrightarrow}N(0,\,V). $$

where variance V is determined by the functional structure of the model.

The result of this theorem follows from Theorem 3.1 in Ai and Chen (2003). A significant difference between (5) used for our estimation and the conditional moment equations implied by infinite-horizon Markov dynamic decision processes is that the one-period-ahead values in our moment equations are estimated separately. As a result, the estimated choice-specific value function and the ex ante value function can be considered to be unrelated nonparametric objects (in contrast to infinite-horizon dynamics, in which the two are connected via a fixed point). This feature facilitates the evaluation of the asymptotic variance.

An explicit expression for the variance can be obtained as follows. We introduce

$$J_{k}(\sigma_{0,t},\sigma_{1,t},\sigma_{0,t+1},\sigma_{1,t+1},s)=\left( \frac{\partial F_{k}}{\partial\sigma_{0,t}},\,\frac{\partial F_{k}} {\partial\sigma_{1,t}},\,\frac{\partial F}{\partial\sigma_{0,t+1}} ,\,\frac{\partial F}{\partial\sigma_{1,t+1}}\right)^{\prime} $$

, J(s)=(J ₁(s), J ₂(s))^′, and

$$\begin{array} [c]{l} M(s)=\\ E\!\left[ \!\!\left. \left( \begin{array} [c]{cccc} \frac{\partial u(s_{t};\theta(1))}{\partial\theta(1)} & 0 & -\frac{\partial u(s_{t};\theta(0))}{\partial\theta(0)}+\beta\frac{\partial u(s_{t+1} ;\theta(0))}{\partial\theta(0)} & F(s_{t+1})+u(s_{t+1};\theta(0))\\ 0 & \frac{\partial u(s_{t};\theta(2))}{\partial\theta(2)} & -\frac{\partial u(s_{t};\theta(0))}{\partial\theta(0)}+\beta\frac{\partial u(s_{t+1} ;\theta(0))}{\partial\theta(0)} & F(s_{t+1})+u(s_{t+1};\theta(0)) \end{array} \right) \right\vert \;s_{t}\,=\,s\!\right] { \!,} \end{array} $$

as well as

$${\Omega}(s)=\text{Var}\left( \left( \widehat{\sigma}_{0,t}(s_{t} ),\,\widehat{\sigma}_{1,t}(s_{t}),\,\widehat{\sigma}_{0,t+1}(s_{t+1} ),\,\widehat{\sigma}_{1,t+1}(s_{t+1})\right) ^{\prime}\,\big\vert\,s_{t} =s\right) . $$

Then, the variance of the second-stage estimates is determined by the sampling noise and the error from the first stage estimates:

$$V=E\left[ M(s_{t})^{-1}\left( E\left[ Z_{t}^{\prime}\text{Var}(\epsilon_{t}|s_{t})Z_{t}\,|\,s_{t}\right] +J(s_{t})\,{\Omega}(s_{t})\,J(s_{t})^{\prime }\right) M(s_{t})^{-1\prime}\,\right] . $$

As an alternative to using the asymptotic formula, we can use the subsampling approach to estimate the variance.

Appendix D: Proof of Theorem 3

In this proof by n we denote the sample size corresponding to the borrowers observed with t periods from mortgage origination. We introduce the notation for the trinomial logit function $\ell (z_{1},z_{2})=\frac {\exp (z_{1})} {1+\exp (z_{1})+\exp (z_{2})}$. By $\tilde {\sigma }_{k,t}^{L}(s)$ we denote the choice probability

$$\tilde{\sigma}_{k,t}^{L}(s)=\ell(\tilde{r}^{L\prime}(t,k)q^{L}(s),\,\tilde {r}^{L\prime}(t,j)q^{L}(s)),\;\;j\neq k $$

where $\tilde {r}^{L}(t,k)$ are the coefficients of the projection of the probability ratio $\log \,\frac {\sigma _{k,t}(s)}{\sigma _{0,t}(s)}$ on L first orthogonal polynomials. We also denote $\tilde {\sigma }_{0,t}^{L} (s)=1-\tilde {\sigma }_{1,t}^{L}(s)-\tilde {\sigma }_{2,t}^{L}(s)$. We note that $\frac {\partial \ell }{\partial z_{1}},\;\frac {\partial \ell }{\partial z_{2}} \leq \frac {1}{2}.$ Thus, $\frac {1}{2}$ is a uniform Lipschitz constant and

$$\begin{array}{@{}rcl@{}} \sup\limits_{s\in{\mathcal{S}}}|\tilde{\sigma}_{k,t}^{L}(s)\!\!\! && -\sigma _{k,t}(s)|=\sup\limits_{s\in{\mathcal{S}}}\left\vert \ell\left( \log \frac{\tilde{\sigma}_{k,t}^{L}(s)}{\tilde{\sigma}_{0,t}^{L}(s)},\,\log \frac{\tilde{\sigma}_{j,t}^{L}(s)}{\tilde{\sigma}_{0,t}^{L}(s)}\right) -\ell\left( \log\frac{\sigma_{k,t}(s)}{\sigma_{0,t}(s)},\,\log\frac {\sigma_{j,t}(s)}{\sigma_{0,t}(s)}\right) \right\vert \\ &&\leq\frac{1}{2}\,\sqrt{\sup\limits_{s\in{\mathcal{S}}}\left\vert \log \frac{\tilde{\sigma}_{1,t}^{L}(s)}{\tilde{\sigma}_{0,t}^{L}(s)}-\log \frac{\sigma_{1,t}(s)}{\sigma_{0,t}(s)}\right\vert^{2}+\sup\limits_{s\in {\mathcal{S}}}\left\vert \log\frac{\tilde{\sigma}_{2,t}^{L}(s)}{\tilde{\sigma }_{0,t}^{L}(s)}-\log\frac{\sigma_{2,t}(s)}{\sigma_{0,t}(s)}\right\vert^{2} }=O(L^{-\alpha}) \end{array} $$

for some $\alpha \geq \frac {1}{2}.$ This guarantees the quality of approximation of the choice probability using a logit transformation of the series expansion.

Now we omit index t in the variables (whenever the period of time under consideration is known), use ${r_{k}^{L}}$ in place of r ^L(t,k), and construct function

$$\begin{array}{@{}rcl@{}} \rho(a,s;{r_{1}^{L}},{r_{2}^{L}})&= & \left( \mathbf{1}\{a=1\}-\mathbf{1} \{a=0\}\right) \ell\left( r_{1}^{L\prime}\,q^{L}(s),\;r_{2}^{L\prime} \,q^{L}(s)\right) \\ && +\left( \mathbf{1}\{a=2\}-\mathbf{1}\{a=0\}\right) \ell\left( r_{2}^{L\prime}\,q^{L}(s),\;r_{1}^{L\prime}\,q^{L}(s)\right) . \end{array} $$

Then we can express the sample quasi-likelihood as

$$\widehat{Q}({r_{1}^{L}},{r_{2}^{L}})=E_{n}\left[ \rho(a,s;{r_{1}^{L}},r_{2}^{L})\right] +E_{n}\left[ \mathbf{1}\{a=0\}\right] , $$

where we adopted the notation from the empirical process theory where $E_{n}[\cdot ]=\frac {1}{n}{\sum }_{i=1}^{n}$. Also introduce the population likelihood with the series expansion

$${Q}({r_{1}^{L}},{r_{2}^{L}})=E\left[ \rho(a,s;{r_{1}^{L}},{r_{2}^{L}})\right] +E\left[ \mathbf{1}\{a=0\}\right] . $$

Consider function

$$\begin{array}{@{}rcl@{}} f(a,s;\,{r_{1}^{L}},{r_{2}^{L}},\tilde{r}_{1}^{L},\tilde{r}_{2}^{L})&=&\rho (a,s;{r_{1}^{L}},{r_{2}^{L}})-\rho(a,s;\tilde{r}_{1}^{L},\tilde{r}_{2}^{L} )-E[\rho(a,s;{r_{1}^{L}},{r_{2}^{L}})]\\&&+E[\rho(a,s;\tilde{r}_{1}^{L},\tilde{r} _{2}^{L})]. \end{array} $$

Provided that we established that function ℓ(⋅,⋅) is Lipschitz, we can evaluate

$$\text{Var}\left( f(a,s;\,{r_{1}^{L}},{r_{2}^{L}},\tilde{r}_{1}^{L},\tilde{r}_{2}^{L})\right) =O\left( L\sup\limits_{k=1,2,\,l\leq L}\Vert r_{k,l} -\tilde{r}_{k,l}\Vert\right) =O(L). $$

where ${r_{k}^{L}}=(r_{k,1},\ldots ,r_{k,L})$. Next we impose a technical assumption that allows us to establish consistency of estimator (4).

Assumption 4

Consider the class of functions indexed by n

$$\begin{array}{@{}rcl@{}} {\mathcal{F}}_{n}&=&\left\{ f(\cdot,\cdot;\,r_{1}^{L_{n}},r_{2}^{L_{n}} ,\tilde{r}_{1}^{L_{n}},\tilde{r}_{2}^{L_{n}})-E[f(\cdot,\cdot;\,r_{1}^{L_{n} },r_{2}^{L_{n}},\tilde{r}_{1}^{L_{n}},\tilde{r}_{2}^{L_{n}})],\;r_{k,l} \in{\Theta},\right.\\&&\left.l\leq L_{n},\,k=1,2\vphantom{\frac{}{}}\right\} , \end{array} $$

where Θ is the compact subset of $\mathbb {R}$ and $\tilde {r}_{1} ^{L_{n}}$ and $\tilde {r}_{2}^{L_{n}}$ are the coefficients of projections of population probability ratios on L _n series terms. Then for each L _n →∞ such that n/(L _n log n)→∞ the L ₁ covering number for class ${\mathcal {F}}_{n}$ , N, has the following bound

$$\log\,N\left( \delta,\,{\mathcal{F}}_{n},\,\mathbf{L}_{1}\right) \leq An^{r_{0}}\log\frac{1}{\delta}, $$

where $0<r_{0}\leq \frac {3}{4}$ and r ₀ ↓0 is assumed to correspond to the factor log n.

This is the condition restricting the complexity of the functions created by logit transformations of series expansions. By construction any $f\in {\mathcal {F}}_{n}$ is bounded |f|<1<∞. We established that Var(f) = O(L _n ) for $f\in {\mathcal {F}}_{n}$. The symmetrization inequality (30) in Pollard (1984) holds if $\varepsilon _{n}/(16n\,\mu _{n}^{2})\leq \frac {1}{2}$. This will occur if $\frac {n\epsilon _{n}}{N^{2} }\rightarrow \infty$. Provided that the symmetrization inequality holds, we can follow the steps of Theorem 37 in Pollard (1984) to establish the tail bound on the deviations of the sample average of f via a combination of the Hoeffding inequality and the covering number for the class ${\mathcal {F}}_{n}$. As a result, we obtain that

$$\begin{array}{@{}rcl@{}} && P\left( \sup\limits_{f\in{\mathcal{F}}_{n}}\frac{1}{n}\biggl\|E_{n} [f(\cdot)]\biggr\|>8\mu_{n}\right) \\ && \leq2\exp\left( An^{r_{0}}\log\frac{1}{\mu_{n}}\right) \exp\left( -\frac{1}{128}\frac{n{\mu_{n}^{2}}}{L_{n}}\right) +P\left( \sup\limits_{f\in {\mathcal{F}}_{n}}\frac{1}{n}\biggl\|E_{n}[f(\cdot)]^{2}\biggr\|>64L_{n} \right) . \end{array} $$

The second term can be evaluated with the aid of Lemma 33 in Pollard (1984):

$$P\left( \sup\limits_{f\in{\mathcal{F}}_{n}}\frac{1}{n}\biggl\|E_{n} [f(\cdot)]^{2}\biggr\|>64L_{n}\right) \leq4\exp\left( An^{2r_{0}}\log \frac{1}{L_{n}}\right) \exp(-nL_{n}). $$

As a result, we find that

$$\begin{array}{@{}rcl@{}} && P\left( \sup\limits_{f\in{\mathcal{F}}_{n}}\frac{1}{n}\biggl\|E_{n} [f(\cdot)]\biggr\|>8\mu_{n}\right) \\ && \leq2\exp\left( An^{r_{0}}\log\frac{1}{\mu_{n}}\right) \exp\left( -\frac{1}{128}\frac{n{\mu_{n}^{2}}}{L_{n}}\right) +4\exp\left( An^{r_{0}} \log\frac{1}{L_{n}}-nL_{n}\right) . \end{array} $$

We start the analysis with the first term. Consider the case with r ₀>0. Then the log of the first term takes the form

$$An^{r_{0}}\log(1/\mu_{n})-\frac{1}{128}\frac{n{\mu_{n}^{2}}}{L_{n}}. $$

Then one needs that $\frac {n{\mu _{n}^{2}}}{L_{n}\,n^{r_{0}}\log \,n} \rightarrow \infty$ if r ₀>0 and $\frac {n{\mu _{n}^{2}}}{L_{n}\,\log ^{2} \,n}\rightarrow \infty$ if r ₀ ↓0. Hence the first term is of o(1). This condition also guarantees that the second term vanishes. We note also that the CLT applies to the term $E_{n}\left [ \mathbf {1}\{a=0\}\right ] =E\left [ \mathbf {1}\{a=0\}\right ] +O_{p}(\frac {1}{\sqrt {n}})$. Now for some slowly diverging sequence δ _n →∞ such that $\mu _{n}=\delta _{n}\sqrt {\frac {L_{n}\,n^{r_{0}}\log \,n}{n}}\rightarrow 0$, we establish that

$$\begin{array}{@{}rcl@{}} \sup\limits_{(r_{1}^{L_{n}},r_{2}^{L_{n}})\in{\Theta}^{L_{n}}\times{\Theta}^{L_{n}}}\!\!\!&&\left\Vert \widehat{Q}({r_{1}^{L}},{r_{2}^{L}})-{Q}({r_{1}^{L}},r_{2} ^{L})+\widehat{Q}(\tilde{r}_{1}^{L},\tilde{r}_{2}^{L})-{Q}(\tilde{r}_{1}^{L},{r_{2}^{L}})\right\Vert \\&&=O_{p}\left( \mu_{n}+\frac{1}{\sqrt{n}}\right) =o_{p}(1). \end{array} $$

Thus, the sample quasi-likelihood converges uniformly to the population quasi-likelihood and the estimated choice probabilities are uniformly consistent over $\mathcal {S}$. To establish the rate for the estimated choice probabilities, we consider a neighborhood of the population projections defined by $\sup \limits _{k=1,2,\,l\leq L_{n}}\Vert r_{k,l}-\tilde {r} _{k,l}\Vert \leq \varepsilon$. Using Lemma 2.3.1 from van der Vaart and Wellner (1996), we can find that

$$E\left[ \sup\limits_{f\in{\mathcal{F}}_{n}}\sqrt{n}E_{n}[f(\cdot)]\right] \leq Cn^{r_{0}/2}\sqrt{L_{n}}\varepsilon\log\frac{1}{\sqrt{L_{n}}\varepsilon }, $$

for some constant C. Using Theorem 3.4.1 from van der Vaart and Wellner (1996) and the derived inequality, we can express the convergence rate for the estimated parameters of the approximated choice probabilities as $\rho _{n}^{2}n^{r_{0}/2}\sqrt {L_{n}}\frac {1}{\rho _{n}}\log \,\frac {\rho _{n}}{\sqrt {L_{n}}}\leq \sqrt {n}$. Then

$$\sup\limits_{s\in{\mathcal{S}}}\left\Vert \widehat{\sigma}_{k,t} (s)-\tilde{\sigma}_{k,t}(s)\right\Vert =O_{p}\left( \frac{L_{n}}{\rho_{n} }\right) . $$

To attain the rate o _p (n ^−1/4) we need to assure that $\frac {L_{n}} {\rho _{n}}=o(n^{-1/4})$. To assure n ^−1/4 we choose δ _n →0 and set L _n = δ _n n ^−1/4 ρ _n . Then the rate constraint can be re-written as

$${\rho_{n}^{2}}n^{-3/8+r_{0}/2}\,\frac{\log\,\frac{n^{1/4}\sqrt{\rho_{n}}} {\sqrt{\delta_{n}}}}{\frac{n^{1/4}\sqrt{\rho_{n}}}{\sqrt{\delta_{n}}}} \leq\sqrt{n}. $$

Provided that $\lim \limits _{x\rightarrow \infty }\log \,x\,/\,x=0$, we conclude that $\rho _{n}=O(n^{7/8-r_{0}/2})$, meaning that $L_{n}=o(n^{5/8-r_{0}/2})$. We note that the slowest rate for the choice of L _n has to satisfy

$$\frac{n^{1-r_{0}}{\mu_{n}^{2}}}{L_{n}\log\,n}\rightarrow\infty, $$

for μ _n →0. Thus, the estimator with rate o(n ^−1/4) is plausible if r ₀<3/4. Using the triangle inequality and our previous result, we find that

$$\sup\limits_{s\in{\mathcal{S}}}\left\Vert \widehat{\sigma}_{k,t}(s)-{\sigma }_{k,t}(s)\right\Vert =O_{p}\left( \frac{L_{n}}{\rho_{n}}+L_{n}^{-\alpha }\right) =o_{p}\left( n^{-1/4}\right) , $$

if α≥1.