Model Adequacy Checks for Discrete Choice Dynamic Models

Abstract

This chapter proposes new parametric model adequacy tests for possibly nonlinear and nonstationary time series models with noncontinuous data distribution, which is often the case in applied work. In particular, we consider the correct specification of parametric conditional distributions in dynamic discrete choice models, not only of some particular conditional characteristics such as moments or symmetry. Knowing the true distribution is important in many circumstances, in particular to apply efficient maximum likelihood methods, obtain consistent estimates of partial effects, and appropriate predictions of the probability of future events. We propose a transformation of data which under the true conditional distribution leads to continuous uniform iid series. The uniformity and serial independence of the new series is then examined simultaneously. The transformation can be considered as an extension of the integral transform tool for noncontinuous data. We derive asymptotic properties of such tests taking into account the parameter estimation effect. Since transformed series are iid we do not require any mixing conditions and asymptotic results illustrate the double simultaneous checking nature of our test. The test statistics converges under the null with a parametric rate to the asymptotic distribution, which is case dependent, hence we justify a parametric bootstrap approximation. The test has power against local alternatives and is consistent. The performance of the new tests is compared with classical specification checks for discrete choice models.

Appendix

Proof of Proposition 1

Part (a) is a property of dynamic PIT with a continuous conditional distribution $F_{t}^{\dagger }$, the proof can be found in Bai (2003). Part (b) follows from the fact that (omitting dependence on $t$, $\Omega _{t}$ and $\theta$)

$$\begin{aligned} F^{\dagger }\left(Y+Z-1\right)&=F\left([Y+Z-1]\right) +Z^U\text{ P}\left([Y+Z]\right) \\&=F\left(Y-1\right) +Z^U\text{ P}\left(Y\right), \end{aligned}$$

where

$$ Z^U=F_z\left(Y+Z-1-[Y+Z-1]\right)=F_z\left(Z\right) $$

is uniform for any $Z\sim F_z$ continuous and with $[0,1]$ support, by the usual static PIT property. Therefore, although a continued variable $Y^{\dagger }$ and its distribution $F^{\dagger }$ depends on $F_z$, $F^{\dagger }(Y^{\dagger })$ does not.

$\square$

Proof of Proposition 2

Assumption 1 in Kheifets (2011) is satisfied automatically after applying continuation defined in (2), therefore Proposition 1 of Kheifets (2011) holds.

$\square$

Proof of Proposition 3

Follows from Kheifets (2011), we need only to check that Assumption 2 in Kheifets (2011) is satisfied.

Let $r =F^{\dagger }\left(y\right).$ Note that $[y]=F^{-1}(r)$ but $F\left([y]\right)=F\left(F^{-1}(r)\right)$ equals $r$ only when $y=[y]$. The inverse of $F^{\dagger }$ is

$$\begin{aligned} y&={\left(F^{\dagger }\right)}^{-1}\left(r\right) =[y]+\frac{r-F\left([y]\right)}{\text{ P}\left([y]+1\right)} =[y]+1+\frac{r-F\left([y]+1\right)}{\text{ P}\left([y]+1\right)}\\&=F^{-1}(r)+\frac{r-F\left(F^{-1}(r)\right)}{\text{ P}\left(F^{-1}(r)+1\right)}. \end{aligned}$$

Note also that $\left(r-F\left([y]\right)\right)/\text{ P}\left([y]+1\right)=y-[y]\in [0,1]$. Take distribution $G$ with the same support as $F$. We have different useful ways to write $d \left(G,F,r\right)$:

$$\begin{aligned} d \left(G,F,r\right)&=\eta ^{\dagger } \left(r\right)-r= G^{\dagger }\left(\left(F^{\dagger }\right)^{-1}\left(r\right)\right)-r =G^{\dagger }\left(y\right)-r \nonumber \\&= G\left(\left[y\right]\right)-F\left(\left[y\right]\right) + \left(y-[y]\right)\left( \text{ P}_G\left(\left[y\right]+1\right)-\text{ P}_F\left(\left[y\right]+1\right)\right)\end{aligned}$$

(A.1)

$$\begin{aligned}&=G\left(\left[y\right]+1\right)-F\left(\left[y\right]+1\right) \nonumber \\&\quad +\left(y-[y]-1\right)\left( \text{ P}_G\left(\left[y\right]+1\right)-\text{ P}_F\left(\left[y\right]+1\right)\right)\end{aligned}$$

(A.2)

$$\begin{aligned}&=G\left(F^{-1}\left(r\right)\right)-F\left(F^{-1}\left(r\right)\right) \nonumber \\&\quad + \frac{r-F\left(F^{-1}(r)\right)}{\text{ P}_F\left(F^{-1}(r)+1\right)}\left( \text{ P}_G\left(F^{-1}(r)+1\right)-\text{ P}_F\left(F^{-1}(r)+1\right)\right). \end{aligned}$$

(A.3)

Thus, noting that $\text{ P}\left(\cdot \right)$ is bounded away from zero, we have that Assumption 2 in this paper is sufficient for Assumption 2 in Kheifets (2011):

• (K2.1)
  $$\begin{aligned} E\sup _{t=1,\ldots ,T}\sup _{u\in B_T}\sup _{r\in [0,1]}\left|\eta ^{\dagger } _{t}\left( r,u,\theta _0\right)-r \right|=O\left( T^{-1/2}\right). \end{aligned}$$

• (K2.2) $\forall M\in (0,\infty )$, $\forall M_2\in (0,\infty )$ and $\forall \delta > 0$

  $$\begin{aligned} \sup _{r\in [0,1]}\frac{1}{\sqrt{T}}\sum _{t=1}^{T}\sup _{\begin{matrix} ||u-v||\le M_2 T^{-1/2-\delta }\\ u,v\in B_T \end{matrix}}\left|\eta ^{\dagger }_{t}\left( r,u,\theta _0\right)-\eta ^{\dagger }_{t}\left(r,v,\theta _0\right) \right|=o_{p}\left(1\right). \end{aligned}$$

• (K2.3) $\forall M\in (0,\infty )$, $\forall M_2\in (0,\infty )$ and $\forall \delta > 0$

  $$\begin{aligned} \sup _{|r-s|\le M_2 T^{-1/2-\delta }}\frac{1}{\sqrt{T}}\sum _{t=1}^{T}\sup _{u\in B_T}\left|\eta ^{\dagger } _{t}\left(r, u,\theta _0\right)-\eta ^{\dagger }_{t}\left(s, u,\theta _0\right)\right|=o_{p}\left(1\right). \end{aligned}$$

• (K2.4) $\forall M\in (0,\infty )$, there exists a uniformly continuous (vector) function $h(r)$ from $[0,1]^{2}$ to $R^{L}$, such that

  $$\begin{aligned} \sup _{u\in B_T }\sup _{r\in [0,1]^{2}}\left|\frac{1}{\sqrt{T}}\sum _{t=2}^{T}h_t-h(r)^{\prime }{\sqrt{T}\left( u-\theta _0\right) } \right|=o_{p}(1). \end{aligned}$$

  where

  $$\begin{aligned} h_t=\left(\eta ^{\dagger } _{t-1}\left( r_{2},u,\theta _0\right) -r_{2}\right) r_{1}+\left( \eta ^{\dagger } _{t}\left( r_{1},u,\theta _0\right) -r_{1}\right) I\left( F^{\dagger } _{t-1}\left( Y^{\dagger } _{t-1}|u\right)\le r_{2}\right). \end{aligned}$$

For Part (a), take $d (F(\cdot |\Omega _{t}, \theta _0),F(\cdot |\Omega _{t}, \hat{\theta }))$. Then (K2.1), (K2.2), (K2.4) follow from (2.1), (2.2) and (2.3) because of representation (A.3). If we compare (A.1) and (A.2) we see that $d(\cdot )$ is not only continuous in $r$, but piece-wise linear, so (K2.3) is satisfied automatically.

For Part (b), take $d (G_T(\cdot |\Omega _{t}, \theta _0),F(\cdot |\Omega _{t}, \hat{\theta }))$ and use the additivity of $d(\cdot )$ in the first arguments:

$$\begin{aligned} d (G_T(\cdot |\Omega _{t}, \theta _0),F(\cdot |\Omega _{t}, \hat{\theta }))&= \left(1-\frac{\sqrt{T_0}}{\sqrt{T}}\right) d (F(\cdot |\Omega _{t}, \theta _0),F(\cdot |\Omega _{t}, \hat{\theta }))\\&\quad +\frac{\sqrt{T_0}}{\sqrt{T}} d (H(\cdot |\Omega _{t}),F(\cdot |\Omega _{t}, \hat{\theta })). \end{aligned}$$

$\square$

Proof of Proposition 5

The proof is similar if we consider $d (F(\cdot |\Omega _{t}, \theta _T),F(\cdot |\Omega _{t}, \hat{\theta }_T))$ under $\{\theta _T:T\ge 1\}$.