Two-step estimation in linear regressions with adaptive learning

Two-step estimation in linear regressions with adaptive learning

\newgeometry{margin=1in}

\restoregeometry

Introduction

Consider a linear regression with adaptive learning

where \(\varepsilon_t\) is an error term, \(\delta_0 \in \mathbb{R}\) and the parameter space of \(\beta_0\) is specified below. The variable \(a_{t} \coloneqq a_t(\theta_0)\) is updated according to a stochastic approximation algorithm

with \(a_0(\theta_0) = \mathfrak{a}_0\) for some initial value \(\mathfrak{a}_0\). The so-called `gain' sequence \(\{\theta_0/t\}_{t\, \geq \,1}\) of positive constants captures the responsiveness to previous prediction mistakes and depends on an unknown gain parameter \(\theta_0\). For \(\theta_0 = 1\), recursion (ref) reduces to recursive least-squares, which updates \(a_t\) by weighing all observations equally. If, in turn, \(\theta_0 > 1\) (\(\theta_0 < 1\)), then earlier observations receive less (more) weight. The dynamics of (ref) with this gain specification have been investigated by masa89 who interpret \(\theta_0\) as a `forgetting factor'; see also nanu15, malmnag16, or malmetal21 for recent economic applications. Typically, more general formulations of the type \(y_t = \beta_0 a_{t-1}x_t + \delta_0x_t + \varepsilon_t\) are considered within this literature, where \(x_t\) is some stochastic regressor and \(y_{t|t-1}^\textnormal{\textsf{e}} \coloneqq a_{t-1}x_t\) is viewed as the subjective expectation an economic agent forms about \(y_t\) at time \(t-1\) by estimating the so-called rational expectations equilibrium \(y_t = \alpha_0x_t + \varepsilon_t\), with \(\alpha_0 \coloneqq \delta_0/(1-\beta_0)\); see evho01. For brevity, we leave this extension with stochastic \(x_t\) to future research.

It is well known from the literature on stochastic approximation that the decreasing-gain specification ensures \(a_t \rightarrow \alpha_0\) in an appropriate probabilistic sense; see bmp90 or lai03 for an introduction to this literature. Although the resulting asymptotic collinearity makes estimation of the model parameters \(\lambda_0 \coloneqq (\beta_0,\delta_0)^\textsf{T}\) difficult, limiting normality and/or consistency of the ordinary least-squares estimator of \(\lambda_0\) can be established under certain regularity conditions; see CM18, CM19, and may21. The major limitation of these papers is, however, that \(\theta_0\) is treated as a known constant. In what follows, I will thus weaken this assumption and treat the case where the gain parameter \(\theta_0\) is first estimated by nonlinear techniques from a noisy sample of \(\{a_t\}_{t \geq 1}\), an exercise commonly encountered in the empirical macroeconomic learning literature where survey data of the subjective expectations \(y_{t|t-1}^\textnormal{\textsf{e}} = a_{t-1}x_t\) are used to infer the unknown gain parameter \(\theta_0\); see, e.g., braev6, mapi14, malmnag16, or bega17. It is, however, so far unclear which (if any) statistical properties any such estimator possesses.

The primary objective of this note is thus to derive weak consistency and asymptotic normality of the prominent nonlinear least-squares estimator of \(\theta_0\). This is achieved by combining classical results from jenn69 and lai94 with more recent arguments due to chawa15 and wang21. In addition, the following section also derives the limiting distribution of the feasible ordinary least-squares estimator of \(\lambda_0\) based on the estimated gain and highlights asymptotic equivalence with its unfeasible counterpart under certain parameter combinations, thereby complementing earlier results from CM18 and may21 (CMM, henceforth). All proofs are contained in the online appendix.

Asymptotic Analysis

Nonlinear Least-Squares Estimation of \(\theta_0\)

Suppose a sample \(\{z_1,\dots,z_n\}\) is observed, where

\(u_t\) is an error term and the notational convention \(a_t = a_t(\theta_0)\) is recalled. This setting corresponds to the empirical macroeconomic learning literature cited above, where \(\{z_1,\dots,z_n\}\) is interpreted as noisy survey data of \(y_{t|t-1}^\textnormal{\textsf{e}} = a_{t-1}x_t\) (with the simplifying assumption \(x_t = 1\)), from which the gain parameter is estimated using the least squares method \[ \hat{\theta}_n \coloneqq \operatorname*{\textsf{arg\,min}}\limits_{\theta \, \in \, [ \underaccent{\bar}{\theta},\bar{\theta}]} Q_n(\theta),\;\;Q_n(\theta) \coloneqq \sum_{t\,=\,1}^n(z_{t}- a_{t-1}(\theta))^2, \quad 0 < \underaccent{\bar}{\theta} \leq \bar{\theta} < \infty. \] Here \(a_{t}(\theta)\) denotes the counterpart of Eq. (ref) that obtains by recursion from Eq. (ref) upon replacing the true \((\theta_0,\mathfrak{a}_0)^\textsf{T}\) with a candidate value \((\theta,\mathfrak{a})^\textsf{T} \in [ \underaccent{\bar}{\theta},\bar{\theta}] \times \mathbb{R}\).\footnote{It is shown in the appendix that the choice of the initial value is asymptotically inconsequential for a suitably restricted parameter space. Thus, in what follows, the dependence of various quantities on the initial value is suppressed.}

Weak Consistency

The consistency result for \(\hat{\theta}_n\) uses a strategy proposed initially by jenn69. Define \(D_n(\theta,\theta_0) \coloneqq Q_n(\theta)-Q_n(\theta_0)\). If, for some sequence \(\nu_n \rightarrow \infty\), \(\nu_n^{-1}D_n(\theta,\theta_0) \stackrel{p}{\rightarrow} D(\theta,\theta_0)\) uniformly in \(\theta \in [ \underaccent{\bar}{\theta},\bar{\theta}]\), where \(\theta \mapsto D(\theta,\theta_0)\) is continuous and has unique minimum \(\theta_0\) $a.s.$, then \(\hat{\theta}_n \stackrel{p}{\rightarrow} \theta_0\). Uniform convergence, in turn, follows from pointwise convergence and a stochastic Lipschitz condition for \(\theta \mapsto Q_n(\theta)\); see, e.g., and92. In order to establish weak consistency of \(\hat{\theta}_n\) along these lines, the following assumptions are imposed.

As a first result, weak consistency is obtained under these assumptions:

Limiting Normality

In order to derive the limiting distribution, the framework of chawa15 is adopted. They show that if

and

holds on \(\Theta_n \coloneqq \{\theta \in [ \underaccent{\bar}{\theta},\bar{\theta}]: \textnormal{\textsf{log}}^{1/2}n|\theta-\theta_0| \leq k_n\}\) for some sequence \(k_n \rightarrow \infty\) as \(n \rightarrow \infty\), then \(\textnormal{\textsf{log}}^{1/2}n(\hat{\theta}_n-\theta_0) = Z_n+o_p(1)\), with \[ Z_n \coloneqq \textnormal{\textsf{log}}^{1/2}n\left[\sum_{t\,=\,1}^n \dot{a}_t^2(\theta_0)\right]^{-1}\sum_{t\,=\,1}^n \dot{a}_t(\theta_0)u_t. \] The latter quantity, in turn, can be shown to converge weakly to a Gaussian random variable. Proposition (ref) summarizes this argument:

Two-Step Estimation of \(\lambda_0\)

Since (ref) is linear in \(\lambda_0\), it is reasonable to estimate \(\lambda_0\) based on the least-squares estimator \(\hat{\lambda}_n(\hat{\theta}_n)\), where \(\hat{\lambda}_n(\theta) \coloneqq \left[\sum_{t\,=\,1}^nw_t(\theta)w_t(\theta)^\textsf{T}\right]^{-1}\sum_{t\,=\,1}^n w_t(\theta)y_t\), with \(w_t(\theta) \coloneqq (1, a_{t-1}(\theta))^\textsf{T}\). To gain some intuition for the following result, it is instructive to note that a second-order Taylor series expansion yields

with \(Z_n\) from Proposition (ref), \(S_n(\theta) \coloneqq \textnormal{\textsf{log}}^{-1/2}n\sum_{t\,=\,1}^n (a_{t-1}(\theta)-\alpha_0)(\varepsilon_t+\beta_0(a_{t-1}(\theta)-a_{t-1}))\), and \(\dot{S}_n(\theta) = \textsf{d}S_n(\theta)/(\textsf{d}\theta)\). Since CMM have shown that \(S_n(\theta_0)\) is asymptotically normal, the following limiting results are mainly due to Proposition (ref) and the fact that the first-order term of the Taylor series expansion obeys

Proposition (ref) formalizes this argument:

Joint Estimation

Let me conclude by briefly commenting on two alternative estimation procedures that do not require an auxiliary model based on a noisy sample like Eq. (ref). More specifically, instead of estimating \(\lambda_0 = (\delta_0,\beta_0)^\textsf{T}\) and \(\theta_0\) sequentially, one could also try to estimate \(\gamma_0 \coloneqq (\delta_0,\beta_0,\theta_0)^\textsf{T}\) jointly based directly on Eq. (ref); see, e.g., mi07, chemama10, or bega17b for empirical applications of joint estimation procedures. In the current context, this would mean to use the following nonlinear least-squares estimator \[ \hat{\gamma}_n \coloneqq \operatorname*{\textsf{arg\,min}}_{\gamma \, \in \, \Gamma} Q_{1,n}(\gamma),\;\; Q_{1,n}(\gamma) \coloneqq \sum_{t\,=\,1}^n(y_t-r_t(\gamma))^2,\;\; r_t(\gamma) \coloneqq \delta+\beta a_{t-1}(\gamma), \] with \(\Gamma \coloneqq [ \underaccent{\bar}{\delta},\bar{\delta}] \times [ \underaccent{\bar}{\beta},\bar{\beta}] \times [ \underaccent{\bar}{\theta},\bar{\theta}]\), where \(-\infty < \underaccent{\bar}{\delta} < \bar{\delta} < \infty\) and \(-\infty < \underaccent{\bar}{\beta} < \bar{\beta} < 1\). This joint estimator fails, however, important sufficient conditions for nonlinear regression. First, there does not exist a sequence \(\{\tau_n\}_{n \geq 1}\), with \(\tau_n \rightarrow \infty\), such that the sample second moment matrix \(M_n(\gamma) \coloneqq \sum_{t\,=\,1}^n\dot{r}_t(\gamma)\dot{r}_t(\gamma)^\textsf{T}\) of the \(3\times 1\) gradient vector \(\dot{r}_t(\gamma) = (1,a_{t-1}(\theta),\beta \dot{a}_{t-1}(\theta))^\textsf{T}\), evaluated at the true parameter values and scaled by \(\tau_n^{-1}\), converges in probability to an invertible matrix, violating, for example, Eq. (2.4) in lai94. Second, jenn69's jenn69 condition for weak consistency is violated, as there does not exist a sequence \(\{\nu_n\}_{n \geq 1}\), with \(\nu_n \rightarrow \infty\), such that \(D_{1,n}(\gamma,\gamma_0) \coloneqq Q_{1,n}(\gamma)-Q_{1,n}(\gamma_0)\) converges, when scaled by \(\nu_n^{-1}\), uniformly in probability to a deterministic function with unique minimum. More specifically, we have on \(\Lambda \coloneqq \{\gamma \in \Gamma: \delta = \alpha_0(1-\beta)\} \subset \Gamma\) \[ \operatorname*{\textnormal{\textsf{sup}}}\limits_{\gamma \, \in \, \Lambda}|\textnormal{\textsf{log}}^{-1}n D_{1,n}(\gamma,\gamma_0)-D_1(\kappa,\kappa_0)| = o_p(1),\;\; \kappa \coloneqq (\beta,\theta)^\textsf{T}, \] where

is a continuous non-negative function of \(\kappa\) that attains its minimum of zero if either (1) \(\kappa = \kappa_0\) or, irrespective of whether \(\theta \neq \theta_0\), (2) \(\beta = \beta_0 = 0\). The non-uniqueness due to case (2) reflects an identification failure resulting from the non-separability of \(\beta\) and \(\theta\) in Eq. (ref). On the other hand, \( \operatorname*{\textnormal{\textsf{sup}}}\limits_{\gamma \, \in \, \Gamma}n^{-1}|D_{1,n}(\gamma,\gamma_0)-\tilde{D}_1(\lambda,\lambda_0)|=o_p(1), \) where \(\tilde{D}_1(\lambda,\lambda_0) \coloneqq (\delta-\alpha_0(1-\beta))^2\), with \(\tilde{D}_1(\lambda,\lambda_0)= 0\) uniformly on \(\Lambda\).\footnote{Interestingly, the estimator of chemama10 is subject to a similar underidentification issue under deacreasing-gain learning; see may21.} Even though these conditions are not necessary, the question of whether weak consistency of \(\hat{\gamma}_n\) or even asymptotic normality can be shown in spite of their violation is beyond the scope of this note.

A little more can be said about the following estimator: Again, the starting point is Eq. (ref), which, after a few rearrangements, yields \(y_t^\star = \beta_0 a_{t-1}^\star + \varepsilon_t\), with \(y_t^\star \coloneqq y_t - \alpha_0\), \(a_t^\star(\theta) \coloneqq a_t(\theta) - \alpha_0\), and \(a_t^\star \coloneqq a_t^\star(\theta_0)\). Treating, for the sake of argument, \(\alpha_0\) as known, we could first estimate \(\kappa_0 = (\beta_0,\theta_0)^\textsf{T}\) using

where \(\Xi \coloneqq [ \underaccent{\bar}{\beta},\bar{\beta}]\times[ \underaccent{\bar}{\theta},\bar{\theta}]\). Contrary to the preceding discussion, the suitably scaled second moment matrix of the \(2 \times 1\) gradient vector \(\dot{\ell}_t(\kappa) = (a_{t-1}^\star(\theta),\beta\dot{a}_{t-1}(\theta))^\textsf{T}\) is positive definite with probability approaching one; i.e. \[ log^{-1}n \sum_{t\,=\,1}^n \dot{\ell}_t(\kappa_0)\dot{\ell}_t(\kappa_0)^T \stackrel{p}{\rightarrow} K \coloneqq

, \] where \(K\) is positive definite as long as \(\beta_0 \neq 0\). Moreover, jenn69's jenn69 condition for weak consistency is satisfied if \(\beta_0 \neq 0\) as \(\textnormal{\textsf{log}}^{-1}n D_{2,n}(\kappa,\kappa_0)\) converges uniformly in probability to \(D_1(\kappa,\kappa_0)\), which, as mentioned previously, is uniquely minimized at \(\kappa_0\) as long as \(\beta_0 \neq 0\). Thus, by ruling out \(\beta_0 = 0\), we obtain from the above and the results contained in the appendix that \( \textnormal{\textsf{log}}^{1/2}n(\hat{\kappa}_n-\kappa_0) \stackrel{d}{\rightarrow} \mathcal{N}(0_2,V_2),\;\; V_2 \coloneqq \sigma_\varepsilon^2K^{-1}. \) Since, by may21, \(\hat{\alpha}_n = \alpha_0 + O_p(n^{-1/2})\), replacing \(\alpha_0\) with \(\hat{\alpha}_n\) in (ref) is asymptotically inconsequential, making \(\hat{\kappa}_n\) a fully feasible estimator. Moreover, in view of \(\alpha_0 = \delta_0/(1-\beta_0)\), an estimator of \(\delta_0\) can be given by \(\hat{\kappa}_{n,\delta} \coloneqq \hat{\alpha}_n(1-\hat{\kappa}_{n,\beta})\), where \(\hat{\kappa}_{n,\beta}\) is the first component of \(\hat{\kappa}_n\). Importantly, if \((1-\beta_0)\theta_0 = 1\), then \(M\) is diagonal and the limiting distributions of both the infeasible and the feasible least-squares estimators from the previous section coincide with that of \(\textnormal{\textsf{log}}^{1/2}n((\hat{\kappa}_{n,\delta},\hat{\kappa}_{n,\beta})^\textsf{T}-\lambda_0)\). Recall that \((1-\beta_0)\theta_0 = 1\) played a crucial role within the context of Proposition (ref). Moreover, the limiting variance of the second component of \(\hat{\kappa}_n\) associated with \(\theta_0\), \(\hat{\kappa}_{n,\theta}\), say, is given by \((2\theta_0-1)(\theta_0(2-\beta_0)-1)^2/(\beta_0\theta_0)^2\), which, contrary to the limiting variance of \(\hat{\theta}_n\) from Proposition (ref), does not depend on innovation variances.

Conclusion

This note derives the singular limiting distribution of the two-step least-squares estimator in a linear regression with estimated gain. Although the limiting distribution is contaminated by the presence of a generated covariate, this generated-regressor problem disappears for certain parameter combinations.

\nolinenumbers

\addcontentsline{toc}{section}{References}

\newgeometry{margin=1.3in}