Inference of Autoregressive Model with Stochastic Exogenous Variable Under Short-Tailed Symmetric Distributions

Abstract

In classical autoregressive models, it is assumed that the disturbances are normally distributed and the exogenous variable is non-stochastic. However, in practice, short-tailed symmetric disturbances occur frequently and exogenous variable is actually stochastic. In this paper, estimation of the parameters in autoregressive models with stochastic exogenous variable and non-normal disturbances both having short-tailed symmetric distribution is considered. This is the first study in this area as known to the authors. In this situation, maximum likelihood estimation technique is problematic and requires numerical solution which may have convergence problems and can cause bias. Besides, statistical properties of the estimators can not be obtained due to non-explicit functions. It is also known that least squares estimation technique yields neither efficient nor robust estimators. Therefore, modified maximum likelihood estimation technique is utilized in this study. It is shown that the estimators are highly efficient, robust to plausible alternatives having different forms of symmetric short-tailedness in the sample and explicit functions of data overcoming the necessity of numerical solution. A real life application is also given.

Appendices

Appendix 1

Proof of asymptotic equivalence of ML and MML

If f(z) is the probability density function of a random variable z, and z _(i) (1 ≤ i ≤ n) are the order statistics of a random sample of size n with t _(i) = E{z _(i)} all finite, then (Hoeffding 1953)

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} g\{ t_{(i)} \} = E\{ g(z)\} = \int_{ - \infty }^{\infty } {g(z)} f(z)\,{\text{d}}z,$$

(19)

where g(z) is a function which satisfies the condition |g(z)| ≤ h(z), h(z) being convex. Since the nonlinear functions $h(u_{i} ) = u_{i} /(1 + [\lambda_{2} /2r_{2} ]u_{i}^{2} )$ and $g(z_{i} ) = z_{i} /(1 + [\lambda_{1} /2r_{1} ]z_{i}^{2} ),$ given in Eq. (6) are all bounded, Eq. (19) is satisfied yielding the asymptotic equivalence of modified likelihood equations and likelihood equations.

Appendix 2

2.1 Sample Information Matrix

The sample information matrix is − 1 times the second derivatives of ln L evaluated at $\mu_{1} = \hat{\mu }_{1} ,$ $\sigma_{1} = \hat{\sigma }_{1} ,$ $\gamma_{1} = \hat{\gamma }_{1} ,$ etc. Asymptotic variances and covariances are obtained from the inverse of this matrix. They provide accurate approximations; see for example the [100,000/n] Monte Carlo run results given in Table 4, where $\gamma_{0} = 1.0,$ $\gamma_{1} = 1.5,$ $\phi = 0.5,$ $\mu_{1} = 0.0$ and $\sigma_{1}$ and $\sigma$ are 1.0. As can be seen the simulated variances given in Table 1 and the ones obtained from sample information matrix are close to each other.

Table 4 Variances of the MML estimators obtained from sample information matrix

Appendix 3

3.1 Asymptotic Variances of the Related Parameters

Write $w_{1i}^{*} = 1 - \lambda_{1} \{ 1 - (\lambda_{1} /2r_{1} )z_{i}^{2} \} /\{ 1 + (\lambda_{1} /2r_{1} )z_{i}^{2} \}^{2}$ and z _i are iid and have the same distribution as z = ε/σ in Eq. (3).

$$E(w_{1i}^{*} ) = \int_{ - \infty }^{\infty } {\left\{ {1 - \lambda_{1} \frac{{1 - (\lambda_{1} /2r_{1} )z^{2} }}{{\left\{ {1 + (\lambda_{1} /2r_{1} )z^{2} } \right\}^{2} }}} \right\}} f(z)\,{\text{d}}z = 1 - \lambda_{1} E(0,2;r_{1} ,\lambda_{1} ) + \frac{{\lambda_{1}^{2} }}{{2r_{1} }}E(1,2;r_{1} ,\lambda_{1} ) = Q$$

since $\frac{1}{{\sqrt {2\pi } }}\int_{ - \infty }^{\infty } {z^{2j} } \exp \left\{ { - \frac{{z^{2} }}{2}} \right\}\,{\text{d}}z = \frac{(2j)!}{{2^{j} (j)!}}$ and

$$E(p,q;r,\lambda ) = E(z^{2p} \{ 1 + (\lambda /2r)\}^{ - q} ) = \frac{{\left[ {\mathop \sum \nolimits_{j = 0}^{r - q} \left( {\begin{array}{*{20}c} {r - q} \\ j \\ \end{array} } \right)\left( {\frac{\lambda }{2r}} \right)^{j} \{ 2(p + j)\} !/2^{p + j} (p + j)!} \right]}}{{\left[ {\mathop \sum \nolimits_{j = 0}^{r} \left( {\begin{array}{*{20}c} r \\ j \\ \end{array} } \right)\left( {\frac{\lambda }{2r}} \right)^{j} \{ 2j\} !/2^{j} j!} \right]}}.$$

It may be noted that $w_{1i}^{*}$ is essentially an increasing function of $z_{i}^{2}$ and is bounded. For large n, $E( z_{(i)} ) \cong t_{(i)} .$ Since variance of $z_{(i)}$ tends to zero as n tends to infinity, the following results (asymptotic) are obtained:

$$\frac{m}{n} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \beta_{1i} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {1 - \lambda_{1} \frac{{1 - (\lambda_{1} /2r_{1} )t_{(i)}^{2} }}{{\left\{ {1 + (\lambda_{1} /2r_{1} )t_{(i)}^{2} } \right\}^{2} }}} \right] \cong E\left( {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} w_{1(i)}^{*} } \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} E(w_{1i}^{*} ) = Q$$

and since u _i and z _i are independent of each other and complete sums are invariant to ordering

$$\bar{u}_{[i]} = \frac{1}{m}\mathop \sum \limits_{i = 1}^{n} \beta_{1i} u_{[i]} \cong \frac{1}{Q}E\left( {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} w_{1(i)}^{*} u_{[i]} } \right) = \frac{1}{Qn}\mathop \sum \limits_{i = 1}^{n} E(w_{1i}^{*} u_{i} ) = \frac{1}{Q}E(w_{1i}^{*} )E(u_{i} ) = E(u_{i} ).$$

Similarly,

$$\begin{aligned} & \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \beta_{1i} \left( {u_{\left[ i \right]} - \bar{u}_{\left[ . \right]} } \right)^{2} \cong E\left[ {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} w_{1\left( i \right)}^{*} \left( {u_{\left[ i \right]} - \bar{u}_{\left[ . \right]} } \right)^{2} } \right] \\ & \quad = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} E\left( {w_{1i}^{*} } \right)E\left( {u_{[i]} - \bar{u}_{[.]} } \right)^{2} = V\left( u \right)Q;\quad V(u) = \mu_{2,2} \\ \end{aligned}$$

and

$$\begin{aligned} & \frac{1}{n}\left\{ {\mathop \sum \limits_{i = 1}^{n} \beta_{1i} y_{[i] - 1}^{2} - \frac{1}{{m_{1} }}\left( {\mathop \sum \limits_{i = 1}^{n} \beta_{1i} y_{[i] - 1} } \right)^{2} } \right\} = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \beta_{1i} \left( {y_{[i] - 1} - \bar{y}_{[i] - 1} } \right)^{2} \\ & \quad \cong E\left[ {\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} w_{1\left( i \right)}^{*} \left( {y_{[i] - 1} - \bar{y}_{[i] - 1} } \right)^{2} } \right] \\ & \quad = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} E\left( {w_{1i}^{*} } \right)E\left( {y_{i - 1} - \bar{y}_{i - 1} } \right)^{2} \\ & \quad = QV(y_{i - 1} ); \quad V(y_{i - 1} ) = \frac{{\gamma_{1}^{2} V(u) + V(\varepsilon )}}{{(1 - \phi^{2} )}} , \\ \end{aligned}$$

where $\bar{y}_{[i] - 1} = \frac{1}{{m_{1} }}\sum\nolimits_{i = 1}^{n} {\beta_{1i} y_{[i] - 1} } .$

More specifically,

$$V(y_{i - 1} ) = V(y_{i} ) = \frac{{\gamma_{1}^{2} V(u) + V(\varepsilon )}}{{(1 - \phi^{2} )}} = \frac{{\gamma_{1}^{2} \mu_{2,2} + \sigma^{2} \mu_{2,1} }}{{(1 - \phi^{2} )}}$$

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since − 1 < ϕ < 1 so that Y is stationary.